FELLER SEMIGROUPS AND MARKOV PROCESSES

Kazuaki TAIRA Department of Mathematics, Hiroshima University, Higashi-Hiroshima 739, Japan

This paper provides a careful and accessible exposition of the functional analytic approach to the problem of construction of Markov processes with boundary conditions in probability theory. Our approach is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. In this paper we construct a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it “dies” at the time when it reaches the set where the particle is definitely absorbed.

Abstra t.

Table of contents 0. Introduction and Results 1. Theory of Feller Semigroups 1.1 Markov Processes 1.2 Markov Transition Functions and Feller Semigroups 1.3 Generation Theorems of Feller Semigroups 2. Theory of Pseudo-Differential Operators 2.1 Function Spaces 2.2 Pseudo-Differential Operators 2.3 Unique Solvability Theorem for Pseudo-Differential Operators 3. Proof of Theorem 1 3.1 General Existence Theorem for Feller Semigroups 3.2 End of Proof of Theorem 1 4. Proof of Theorem 2 4.1 The Space C0 (D \ M ) 4.2 End of Proof of Theorem 2 Appendix: The maximum principle References 1991 Mathematics Subject Classification. Primary 47D07, 35J25; Secondary 47D05, 60J35, 60J60. Key words and phrases. Feller semigroup, Markov process, elliptic boundary value problem. Typeset by AMS-TEX 1

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KAZUAKI TAIRA

0. Introduction and Results Let D be a bounded, convex domain of Euclidean space RN with smooth boundary ∂D; its closure D = D ∪ ∂D is an N -dimensional compact smooth manifold with boundary. ∂D ...........................

................................................. .......... . ... ......•.... . . . . ..... ... . . . ...... . ... .... ..... . . .. .. . ..... ...... . ... n D .. ... . ..... ... . . ....... . . .......... ....... . . . . . . . ...................... . . . ............................................ Figure 0.1 Let C(D) be the space of real-valued, continuous functions on D. We equip the space C(D) with the topology of uniform convergence on the whole D; hence it is a Banach space with the maximum norm kf k∞ = max |f (x)|. x∈D

A strongly continuous semigroup {Tt }t≥0 on the space C(D) is called a Feller semigroup on D if it is non-negative and contractive on C(D): f ∈ C(D), 0 ≤ f ≤ 1

on D =⇒ 0 ≤ Tt f ≤ 1 on D.

It is known (see [Ta1]) that if Tt is a Feller semigroup on D, then there exists a unique Markov transition function pt on D such that Z pt (x, dy)f (y), f ∈ C(D). Tt f (x) = D

It can be shown that the function pt is the transition function of some strong Markov process; hence the value pt (x, E) expresses the transition probability that a Markovian particle starting at position x will be found in the set E at time t. Furthermore, it is known (see [BCP], [SU], [Ta1], [We]) that the infinitesimal generator of a Feller semigroup {Tt }t≥0 is described analytically by a Waldenfels operator W and a Ventcel’ boundary condition L, which we formulate precisely. Let W be a second-order elliptic integro-differential operator with real coefficients such that (0.1)

W u(x) = P u(x) + Sr u(x) N N 2 X X ∂u ∂ u (x) + bi (x) (x) + c(x)u(x) := aij (x) ∂x ∂x ∂x i j i i=1 i,j=1 Z N X ∂u + s(x, y) u(y) − σ(x, y) u(x) − (x) dy. (yj − xj ) ∂x j D j=1

FELLER SEMIGROUPS AND MARKOV PROCESSES

3

Here: (1) aij ∈ C ∞ (RN ), aij (x) = aji (x) for x ∈ boldRN and 1 ≤ i, j ≤ N , and there exists a constant a0 > 0 such that N X

i,j=1

aij (x)ξi ξj ≥ a0 |ξ|2 ,

x ∈ RN , ξ ∈ RN .

(2) bi ∈ C ∞ (RN ). (3) c ∈ C ∞ (RN ) and c(x) ≤ 0 in D. (4) The integral kernel s(x, y) is the distribution kernel of a properly supported, N pseudo-differential operator S ∈ L2−κ 1,0 (R ), κ > 0, which has the transmission property with respect to ∂D (see Subsection 2.2), and s(x, y) ≥ 0 off the diagonal {(x, x) : x ∈ RN } in RN × RN . The measure dy is the Lebesgue measure on RN . (5) The function σ(x, y) is a local unity function on D, that is, σ(x, y) is a smooth function on D × D such that σ(x, y) = 1 in a neighborhood of the diagonal ∆D = {(x, x) : x ∈ D} in D × D. The function σ(x, y) depends on the shape of the domain D. More precisely, it depends on a family of local charts on D in each of which the Taylor expansion is valid for functions u. For example, if D is convex , we may take σ(x, y) ≡ 1 on D × D. R (6) W 1(x) = P 1(x) + Sr 1(x) = c(x) + D s(x, y) [1 − σ(x, y)] dy ≤ 0 in D. The intuitive meaning of condition (6) is that the jump phenomenon from a point x ∈ D to the outside of a neighborhood of x in the interior D is “dominated” by the absorption phenomenon at x. In particular, if c(x) ≡ 0 in D, then condition (6) implies that any Markovian particle does not move by jumps from x ∈ D to the outside of a neighborhood V (x) of x in the interior D, since we have the assertion Z

D

s(x, y) [1 − σ(x, y)] dy = 0,

and so, by conditions (4) and (5), s(x, y) = 0 for all y ∈ D \ V (x).

.......................................... ................. ......... . . . . . . . D ..... ...... . ..... . . ....... . .... .... .... . . .. . ... ... . .. . ... ... .. . . . . .. . ..... . .. ... . . ... . . . . . ..... .... ... . . . . . ... ... .. . . . ... ....... .. ....... ... ........ ... . . . . . . . . . ... ..... ... ....... ... ... ... . ... ... ...... .... .... .. .. ... .... . . . ...... . ..... ....... . . . . . . .......... ............................................................. Figure 0.2

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KAZUAKI TAIRA

The operator W is called a second-order Waldenfels operator (cf. [BCP]). The differential operator P is called a diffusion operator which describes analytically a strong Markov process with continuous paths (diffusion process) in the interior D. The operator Sr is called a second-order L´evy operator which is supposed to correspond to the jump phenomenon in the interior D; a Markovian particle moves by jumps to a random point, chosen with kernel s(x, y), in the interior D. Therefore, the Waldenfels operator W is supposed to correspond to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space D (see Figure 0.2). It should be emphasized that the integral operator Sr is a “regularization” of S, since the integrand is absolutely convergent. Indeed, we can write Sr u(x) in the form Sr u(x) Z N X ∂u (x) dy = s(x, y) u(y) − σ(x, y) u(x) + (yj − xj ) ∂x j D j=1 Z = s(x, y) [1 − σ(x, y)] u(y) dy D Z N X ∂u (x) dy. + s(x, y)σ(x, y) u(y) − u(x) − (yj − xj ) ∂x j D j=1

By using Taylor’s formula u(y) − u(x) − =

N X

i,j=1

N X j=1

(yj − xj )

(yi − xi )(yj − xj )

Z

∂u (x) ∂xj

0

1

∂ 2u (x + t(y − x))dt , (1 − t) ∂xi ∂xj

we can find a constant C1 > 0 such that N X ∂u ≤ C1 |x − y|2 , u(y) − u(x) − (x) (y − x ) j j ∂x j j=1

x, y ∈ D.

On the other hand, it follows from an application of [CM, Chapitre IV, Proposition 1]) that, for any compact K ⊂ RN , there exists a constant C2 > 0 such that the N distribution kernel s(x, y) of S ∈ L2−κ 1,0 (R ), κ > 0, satisfies the estimate 0 ≤ s(x, y) ≤

C2 , |x − y|N+2−κ

x, y ∈ D, x 6= y.

Therefore, we have, with some constant C3 > 0, Z N X ∂u s(x, y)σ(x, y) u(y) − u(x) − (yj − xj ) (x) dy ∂xj D j=1

FELLER SEMIGROUPS AND MARKOV PROCESSES

≤ C3 kukC 2 (D) = C3 kukC 2 (D)

Z

ZD D

5

1 · |x − y|2 dy |x − y|N+2−κ 1 dy. |x − y|N−κ

Similarly, we have, with some constant C4 > 0, Z Z s(x, y) [1 − σ(x, y)] u(y) dy ≤ C4 kuk C(D) D

D

1 dy, |x − y|N−κ

since we have the formula σ(x, y) − 1

= σ(x, y) − σ(x, x) − =

N X

i,j=1

N X j=1

(yi − xi )(yj − xj )

(yj − xj )

Z

1 0

∂σ (x, x) ∂xj

∂ 2σ (1 − t) (x, x + t(y − x))dt . ∂xi ∂xj

Let L be a second-order boundary condition such that in local coordinates (x1 , x2 , . . . , xN−1 ) (0.2) ∂u Lu(x′ ) = Qu(x′ ) + µ(x′ ) (x′ ) − δ(x′ )W u(x′ ) + Γr u(x′ ) ∂n N−1 N−1 2 X X ∂ u ∂u ′ αij (x′ ) := β i (x′ ) (x′ ) + (x ) + γ(x′ )u(x′ ) ∂x ∂x ∂x i j i i,j=1 i=1

∂u + µ(x′ ) (x′ ) − δ(x′ )W u(x′ ) ∂n Z N−1 X ∂u ′ ′ (x ) dy + r(x′ , y ′ ) u(y ′ ) − τ (x′ , y ′ ) u(x′ ) − (yj − xj ) ∂x j ∂D j=1 Z + t(x′ , y) [u(y) − u(x′ )] dy . D

Here: (1) The operator Q is a second-order degenerate elliptic differential operator on ij ∂D with non-positive principal symbol. In other words the α are the compo2 nents of a smooth symmetric contravariant tensor of type 0 on ∂D satisfying the condition N−1 X

i,j=1

ij

′

α (x )ξi ξj ≥ 0,

′

′

x ∈ ∂D, ξ =

N−1 X

Here Tx∗′ (∂D) is the cotangent space of ∂D at x′ .

j=1

ξj dxj ∈ Tx∗′ (∂D).

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KAZUAKI TAIRA

PN−1 (2) β(x′ ) = i=1 β i (x′ )∂u/∂xi is a smooth vector field on ∂D. (3) Q1 = γ ∈ C ∞ (∂D) and γ(x′ ) ≤ 0 on ∂D. (4) µ ∈ C ∞ (∂D) and µ(x′ ) ≥ 0 on ∂D. (5) δ ∈ C ∞ (∂D) and δ(x′ ) ≥ 0 on ∂D. (6) n = (n1 , n2 , . . . , nN ) is the unit interior normal to the boundary ∂D. (7) The integral kernel r(x′ , y ′ ) is the distribution kernel of a pseudo-differential 2−κ1 operator R ∈ L1,0 (∂D), κ1 > 0, and r(x′ , y ′ ) ≥ 0 off the diagonal ∆∂D = ′ ′ ′ {(x , x ) : x ∈ ∂D} in ∂D × ∂D. The density dy ′ is a strictly positive density on ∂D. (8) The integral kernel t(x, y) is the distribution kernel of a properly supported, 1−κ2 (RN ), κ2 > 0, which has the transmission pseudo-differential operator T ∈ L1,0 property with respect to the boundary ∂D, and t(x, y) ≥ 0 off the diagonal {(x, x) : x ∈ RN } in RN × RN . (9) The function τ (x, y) is a local unity function on D; more precisely, τ (x, y) is a smooth function on D ×D, with compact support in a neighborhood of the diagonal ∆∂D , such that, at each point x′ of ∂D, τ (x′ , y) = 1 for y in a neighborhood of x′ in D. The function τ (x, y) depends on the shape of the boundary ∂D. (10) The operator Γr is a boundary condition of order 2 − κ1 , and satisfies the condition Z ′ ′ ′ Q1(x ) + Γr 1(x ) = γ(x ) + r(x′ , y ′ ) [1 − τ (x′ , y ′ )] dy ′ ≤ 0 on ∂D. ∂D

The intuitive meaning of condition (10) is that the jump phenomenon from a point x′ ∈ ∂D to the outside of a neighborhood of x′ on the boundary ∂D is “dominated” by the absorption phenomenon at x′ . In particular, if γ(x′ ) ≡ 0 on ∂D, then condition (10) implies that any Markovian particle does not move by jumps from x′ ∈ ∂D to the outside of a neighborhood V (x′ ) of x′ on the boundary ∂D, since we have the assertion Z r(x′ , y ′ ) [1 − τ (x′ , y ′ )] dy ′ = 0, ∂D

and so, by conditions (7) and (9), r(x′ , y ′ ) = 0 for all y ′ ∈ ∂D \ V (x′ ). It should be noticed that the integral operator Γr u(x′ ) Z N−1 X ∂u ′ ′ (x ) dy (yj − xj ) = r(x′ , y ′ ) u(y ′ ) − τ (x′ , y ′ ) u(x′ ) + ∂xj ∂D j=1 Z + t(x′ , y) [u(y) − u(x′ )] dy, x′ ∈ ∂D, D

2−κ1 1−κ2 is a “regularization” of R ∈ L1,0 (∂D) and T ∈ L1,0 (RN ), since the integrals N−1 X ′ ∂u ′ ′ ′ ′ ′ (yj − xj ) Rr u(x ) = r(x , y ) u(y ) − τ (x , y ) u(x ) + (x ) dy , ∂xj ∂D j=1 ′

Z

′

′

FELLER SEMIGROUPS AND MARKOV PROCESSES ′

Tr u(x ) =

Z

D

7

t(x′ , y) [u(y) − u(x′ )] dy

are both absolutely convergent. Indeed, it suffices to note that the kernels r(x′ , y ′ ) of R and t(x′ , y) of T satisfiy respectively the estimates C′

, x′ , y ′ ∈ ∂D, x′ 6= y ′ , |x′ − y ′ |(N−1)+2−κ1 C ′′ ′ , x′ ∈ ∂D, y ∈ D, 0 ≤ t(x , y) ≤ ′ N+1−κ 2 |x − y|

0 ≤ r(x′ , y ′ ) ≤

where |x′ − y ′ | denotes the geodesic distance between x′ and y ′ with respect to the Riemannian metric of the manifold ∂D. The boundary condition L is called a second-order Ventcel’ boundary condition (cf. [We]). The six terms of L N−1 X

i,j=1

N−1 X ∂ 2u ∂u ′ ′ α (x ) (x ) + β i (x′ ) (x ), ∂xi ∂xj ∂xi ij

′

i=1

γ(x′ )u(x′ ),

∂u µ(x′ ) (x′ ), ∂n

δ(x′ )W u(x′ ), N−1 X

∂u ′ ′ (x ) dy , ∂xj ∂D j=1 Z N−1 X ∂u ′ (x ) dy t(x′ , y) u(y) − u(x′ ) − (yj − xj ) ∂xj D j=1 Z

r(x′ , y ′ ) u(y ′ ) − u(x′ ) −

(yj − xj )

are supposed to correspond to the diffusion along the boundary, the absorption phenomenon, the reflection phenomenon, the viscosity phenomenon and the jump phenomenon on the boundary and the inward jump phenomenon from the boundary, respectively (see Figures 0.3 through 0.5 below). This paper is devoted to the functional analytic approach to the problem of construction of Feller semigroups with Ventcel’ boundary conditions. More precisely, we consider the following problem: Problem. Conversely, given analytic data (W, L), can we construct a Feller semigroup {Tt }t≥0 whose infinitesimal generator is characterized by (W, L) ?

... ... D ... .. .. ..... ... ..... ... . ... . . . . ... ... .. ... . ... . . . . ... ....... ....... ...... .. .. . . ..... . . . . . ∂D ....... ...... .................... .... . . . ....... ............ . ................................................

... ... D ... ... ... .. ..... .... ... . ..... ....... . .... ... ... ... .. .. .. ... ... . ... .. . . . . . . .. ...... ... .. ... . ∂D ....... ...................... ... . . . . . . . ...... ............ ................ ..... ....... ........... ...... ...... .............................................

absorption

reflection Figure 0.3

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KAZUAKI TAIRA

... ... ... ... ... D ... ... . . ... .. ... . .. .. . ∂D ....... . ...... .... . . ....... . . . ................................................

... ... ... ... ... .. D ... . . . ... . ... .. . .... .... . . . ..... . ∂D ..... ... .. ...... ... ...... ..... ... . . .............. ................ ........ .............. ....... .................. ........................................................

diffusion along the boundary

viscosity

Figure 0.4

... ... ... ... ... .. D . ... . . ... . . . ....... ... .. ... .. . . .... . . . ∂D ..... ....... ... . ...... .... . . .. ......... ....................................

... ... ... .. ... .. D . ... . . ... . . ... .. .... . ∂D ..... .. . . ...... . . . ................................................................... ..............................

jump into the interior

jump on the boundary Figure 0.5

We shall only restrict ourselves to some aspects which have been discussed in our papers [Ta1] through [Ta4]. Our approach is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. It focuses on the relationship between two interrelated subjects in analysis; Feller semigroups and elliptic boundary value problems, providing powerful methods for future research. Now we say that the boundary condition L is transversal on the boundary ∂D if it satisfies the condition Z (0.3) t(x′ , y)dy = +∞ if µ(x′ ) = δ(x′ ) = 0. D

The intuitive meaning of condition (0.3) is that a Markovian particle jumps away “instantaneously” from the points x′ ∈ ∂D where neither reflection nor viscosity phenomenon occurs (which is similar to the reflection phenomenon). Probabilistically, this means that every Markov process on the boundary ∂D is the “trace” on ∂D of trajectories of some Markov process on the closure D = D ∪ ∂D. The next theorem asserts that there exists a Feller semigroup on D corresponding to such a diffusion phenomenon that one of the reflection phenomenon, the viscosity phenomenon and the inward jump phenomenon from the boundary occurs at each point of the boundary ∂D (cf. [Ta4, Theorem 1]): Theorem 1. We define a linear operator A from the space C(D) into itself as follows:

FELLER SEMIGROUPS AND MARKOV PROCESSES

9

(a) The domain of definition D(A) of A is the set (0.4) D(A) = u ∈ C(D) : W u ∈ C(D), Lu = 0 .

(b) Au = W u for all u ∈ D(A). Here W u and Lu are taken in the sense of distributions. Assume that the boundary condition L is transversal on the boundary ∂D. Then the operator A generates a Feller semigroup {Tt }t≥0 on D. The situation may be represented schematically by Figure 0.6 below.

............ ....................... ................................ . . . . . . . . ....... ∂D...... ...... ... . . . .... ..... ... .. ... .. ... .. .. ... D .. ... ... . . ... ..... .. ... ...... .. . . . . . . . . . . . . ......... .................................... ........ ... .... .. ... .......... .......... .. ........ ........ .. . . . . . ....... ..... ... .... .... ... .... ......... ......... ........ ... ... ..... . . . . . . . ......... . ......................... ........................................... Figure 0.6 It should be noticed that Theorem 1 was proved before by Taira [Ta1, Theorem 10.1.3] under some additional conditions, and also by Cancelier [Ca, Th´eor`eme 3.2]. On the other hand, Takanobu and Watanabe [TW] proved a probabilistic version of Theorem 1 in the case where the domain D is the half space RN + (see [TW, Corollary]). Next we generalize Theorem 1 to the non-transversal case. To do this, we assume that (H) There exists a second-order Ventcel’ boundary condition Lν such that Lu = m(x′ ) Lν u + γ(x′ ) u on ∂D, where (3′ ) m ∈ C ∞ (∂D) and m(x′ ) ≥ 0 on ∂D, and the boundary condition Lν is given in local coordinates (x1 , x2 , . . . , xN−1 ) by the formula Lν u(x′ ) ∂u = Qu(x′ ) + µ(x′ ) (x′ ) − δ(x′ )W u(x′ ) + Γu(x′ ) ∂n N−1 N−1 2 X X ∂ u ∂u ′ i αij (x′ ) β (x′ ) := (x′ ) + (x ) ∂x ∂x ∂x i j i i,j=1 i=1

10

KAZUAKI TAIRA

∂u + µ(x′ ) (x′ ) − δ(x′ ) W u(x′ ) ∂n Z N−1 X ∂u ′ ′ r(x′ , y ′ ) u(y ′ ) − u(x′ ) − (yj − xj ) + (x ) dy ∂xj ∂D j=1 Z N−1 X ∂u ′ t(x′ , y) u(y) − u(x′ ) − (yj − xj ) (x ) dy , + ∂x j D j=1

and satisfies the transversality condition Z ′ t(x′ , y) dy = +∞ if µ(x′ ) = δ(x′ ) = 0. (0.3 ) D

We let

′

′

′

M = {x ∈ ∂D : µ(x ) = δ(x ) = 0, Then, by condition (0.3′ ) it follows that

Z

D

t(x′ , y) dy < ∞}.

M = {x′ ∈ ∂D : m(x′ ) = 0},

since we have µ(x′ ) = m(x′ ) µ(x′ ), δ(x′ ) = m(x′ ) δ(x′ ), and t(x′ , y) = m(x′ ) t(x′ , y). Hence we find that the boundary condition L is not transversal on ∂D. Furthermore, we assume that (A) m(x′ ) − γ(x′ ) > 0 on ∂D. The intuitive meaning of conditions (H) and (A) is that a Markovian particle does not stay on ∂D for any period of time until it “dies” at the time when it reaches the set M where the particle is definitely absorbed. Now we introduce a subspace of C(D) which is associated with the boundary condition L. By condition (A), we find that the boundary condition Lu = m(x′ ) Lν u + γ(x′ ) u = 0 on ∂D includes the condition u = 0 on M . With this fact in mind, we let C0 (D \ M ) = {u ∈ C(D) : u = 0 on M }.

The space C0 (D \ M ) is a closed subspace of C(D); hence it is a Banach space. A strongly continuous semigroup {Tt }t≥0 on the space C0 (D \ M ) is called a Feller semigroup on D \ M if it is non-negative and contractive on C0 (D \ M ): f ∈ C0 (D \ M ), 0 ≤ f ≤ 1 on D \ M =⇒ 0 ≤ Tt f ≤ 1 on D \ M .

We define a linear operator W from C0 (D \ M ) into itself as follows: (a) The domain of definition D(W) of W is the set (0.5) D(W) = u ∈ C0 (D \ M ) : W u ∈ C0 (D \ M ), Lu = 0 .

(b) Wu = W u for all u ∈ D(W). The next theorem is a generalization of Theorem 1 to the non-transversal case (cf. [Ta4, Theorem 2]):

FELLER SEMIGROUPS AND MARKOV PROCESSES

11

Theorem 2. Assume that conditions (A) and (H) are satisfied. Then the operator W defined by formula (0.5) generates a Feller semigroup {Tt }t≥0 on D \ M . If Tt is a Feller semigroup on D \M , then there exists a unique Markov transition function pt on D \ M such that Tt f (x) =

Z

pt (x, dy)f (y),

D\M

f ∈ C0 (D \ M ),

and further that pt is the transition function of some strong Markov process. On the other hand, the intuitive meaning of conditions (A) and (H) is that the absorption phenomenon occurs at each point of the set M = {x′ ∈ ∂D : m(x′ ) = 0}. Therefore, Theorem 2 asserts that there exists a Feller semigroup on D \ M corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space D \ M until it “dies” at the time when it reaches the set M . The situation may be represented schematically by Figure 0.7. ...................................................................... ............... ............ ........... ......... . . . . . . . ....... ∂D .......... ..... .. . ..... . . . . . .... . . . ... . . ... . . . .. ... ........... . . . .... . ... .. . . . . . .. . ... . .. . . . . D .. ...... . . ... ... ... . ... ..... .. . . . ... .. . .. ..... ....... .. .. ...... .............. ... ...... ... . . ... .................................. . . ... ........ ..... ....... ..... ...... ....... ..... ........ . . . . . . . . . . . . .......... ....... ..... ............. ..... .......... ....... .......

...... . . . . . . . . . . . . . . . ................................. M

Figure 0.7 It should be noticed that Taira [Ta2] has proved Theorem 2 under the condition that Lν = ∂/∂n and δ(x′ ) ≡ 0 on ∂D, by using the Lp theory of pseudo-differential operators (see [Ta2, Theorem 4]). Finally, we consider the case where all the operators S, T and R are pseudodifferential operators of order less than one. Then we can take σ(x, y) ≡ 1 on D × D, and write the operator W in the following form: (0.1′ )

W u(x) = P u(x) + Sr u(x) X N N X ∂ 2u ∂u ij i := a (x) b (x) (x) + (x) + c(x)u(x) ∂xi ∂xj ∂xi i,j=1 i=1 Z + s(x, y)[u(y) − u(x)]dy , D

where: (4′ ) The integral kernel s(x, y) is the distribution kernel of a properly supported, N pseudo-differential operator S ∈ L1−κ 1,0 (R ), κ > 0, which has the transmission

12

KAZUAKI TAIRA

property with respect to the boundary ∂D, and s(x, y) ≥ 0 off the diagonal {(x, x) : x ∈ RN } in RN × RN . (6′ ) W 1(x) = c(x) ≤ 0 in D. Similarly, the boundary condition L can be written in the following form: ∂u ′ (x ) − δ(x′ )W u(x′ ) + Γu(x′ ) ∂n N−1 N−1 X X ∂ 2u ij ′ i ′ ∂u ′ ′ ′ ′ α (x ) β (x ) := (x ) + (x ) + γ(x )u(x ) ∂x ∂x ∂x i j i i,j=1 i=1

(0.2′ ) Lu(x′ ) = Qu(x′ ) + µ(x′ )

N−1 X ∂u ′ ∂u ′ ′ ′ ′ ′ + µ(x ) (x ) − δ(x )W u(x ) + η(x )u(x ) + ζ i (x′ ) (x ) ∂n ∂x i i=1 Z Z ′ ′ ′ ′ ′ ′ ′ + r(x , y )[u(y ) − u(x )]dy + t(x , y)[u(y) − u(x )]dy , ′

∂D

D

where: (6′ ) The integral kernel r(x′ , y ′ ) is the distribution kernel of a pseudo-differential 1−κ1 operator R ∈ L1,0 (∂D), κ1 > 0, and r(x′ , y ′ ) ≥ 0 off the diagonal {(x′ , x′ ) : x′ ∈ ∂D} in ∂D × ∂D. (7′ ) The integral kernel t(x, y) is the distribution kernel of a properly supported, 1−κ2 pseudo-differential operator T ∈ L1,0 (RN ), κ2 > 0, which has the transmission property with respect to the boundary ∂D, and t(x, y) ≥ 0 off the diagonal {(x, x) : x ∈ RN } in RN × RN . (9′ ) Γ1(x′ ) = η(x′ ) ≤ 0 on ∂D. Then Theorems 1 and 2 may be simplified as follows:

Theorem 3. Assume that the operator W and the boundary condition L are of the forms (0.1′ ) and (0.2′ ), respectively. If the boundary condition L is transversal on the boundary ∂D, then the operator W defined by formula (0.4) generates a Feller semigroup {Tt }t≥0 on D. Theorem 4. Assume that the operator W and the boundary condition L are of the forms (0.1′ ) and (0.2′ ), respectively. If conditions (A) and (H) are satisfied, then the operator W defined by formula (0.6) generates a Feller semigroup {Tt }t≥0 on D \ M. Theorems 1, 2, 3 and 4 solve from the viewpoint of functional analysis the problem of construction of Feller semigroups with Ventcel’ boundary conditions for elliptic Waldenfels operators. The rest of this paper is organized as follows. In Section 1 we present a brief description of basic definitions and results about a class of semigroups (Feller semigroups) associated with Markov processes, which forms a functional analytic background for the proof of Theorems 1 and 2. Section 2 provides a review of basic concepts and results of the theory of pseudodifferential operators – a modern theory of potentials – which will be used in the subsequent sections. In particular we introduce the notion of transmission property due to Boutet de Monvel [Bo], which is a condition about symbols in the normal

FELLER SEMIGROUPS AND MARKOV PROCESSES

13

direction at the boundary. Furthermore, we prove an existence and uniqueness theorem for a class of pseudo-differential operators which enters naturally in the construction of Feller semigroups. Section 3 is devoted to the proof of Theorem 1. We reduce the problem of construction of Feller semigroups to the problem of unique solvability for the boundary value problem (α − W )u = f in D, (λ − L)u = ϕ

on ∂D,

v=0

on ∂D.

and then prove existence theorems for Feller semigroups. Here α is a positive number and λ is a non-negative number. The idea of our approach is stated as follows (cf. [BCP], [RS], [Ta1]). First, we consider the following Dirichlet problem: (α − W )v = f in D, The existence and uniqueness theorem for this problem is well established in the framework of H¨older spaces. We let v = G0α f. The operator G0α is the Green operator for the Dirichlet problem. Then it follows that a function u is a solution of the problem (α − W )u = f in D, (∗) Lu = 0 on ∂D if and only if the function w = u − v is a solution of the problem (α − W )w = 0 in D, Lw = −Lv = −LG0α f

on ∂D.

However, we know that every solution w of the equation (α − W )w = 0 in D can be expressed by means of a single layer potential as follows: w = Hα ψ. The operator Hα is the harmonic operator for the Dirichlet problem. Thus, by using the Green and harmonic operators we can reduce the study of problem (∗) to that of the equation LHα ψ = −LG0α f on ∂D. This is a generalization of the classical Fredholm integral equation. It is known (see [Bo], [Ho], [RS]) that the operator LHα is a pseudo-differential operator of second order on the boundary ∂D.

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KAZUAKI TAIRA

By using the H¨older space theory of pseudo-differential operators, we can show that if the boundary condition L is transversal on the boundary ∂D, then the operator LHα is bijective in the framework of H¨older spaces. The crucial point in the proof is that we consider the term δ(x′ )(W u|∂D ) of viscosity in the boundary condition Lu = L0 u − δ(x′ )(W u|∂D ) as a term of “perturbation” of the boundary condition L0 u. Therefore, we find that a unique solution u of problem (∗) can be expressed in the form u = G0α f − Hα LHα−1 LG0α f . This formula allows us to verify all the conditions of the generation theorems of Feller semigroups discussed in Subsection 1.2. Intuitively this formula tells us that if the boundary condition L is transversal on the boundary ∂D, then we can “piece together” a Markov process on the boundary ∂D with W -diffusion in the interior D to construct a Markov process on the closure D = D ∪ ∂D. The situation may be represented schematically by Figure 0.8.

....................................... .................. ................ . . . . . . . ........ .. ............. . ..... . . . . . . . . . . ..... ....... .................. . . ............ ... . . .... ... ... ..... ∂D ..... . . . ............ . . . .... ... ..... .. ..... ... . .. . . . .. . .. ........ ...... ... ..... ...... . ... . ... . . .. . ..... . . . . . . . ..... ..... .... ... . . ....... . . . ................................................ ................... .... ....... .................. ... .. . .......... ....... ... ... ......... . . .. . . . . ..... ..... . D ......................... ... .......... . . . ....... ..... ........ ..... ... ..... ......... ..... .. ... ......... . . . ... . . . . . .. ....... . .......... ... . . . . . . ....... ... ... .... .................. ........ ........... ... .... .. .. ..... ......... ..... . .... .... . . . . . ..... .... ...... .... ......... ... ..... ..... . ....... . ..... .................... . . . . . . . ..... . ......... . ...... . .............................................. . . . . . . . . ..................................... Figure 0.8 It seems that our method of construction of Feller semigroups is, in spirit, not far removed from the probabilistic method of construction of diffusion processes by means of Poisson point processes of Brownian excursions used by Watanabe [Wa]. In Section 4 we prove Theorem 2. We explain the idea of the proof. First, we remark that if condition (H) is satisfied, then the boundary condition L can be written in the form Lu = m(x′ ) Lν u + γ(x′ ) u on ∂D, where the boundary condition Lν is transversal on ∂D. Hence, by applying Theorem 1 to the boundary condition Lν we can solve uniquely the following boundary value problem: (α − W )v = f in D, Lν v = 0

on ∂D.

FELLER SEMIGROUPS AND MARKOV PROCESSES

15

We let v = Gνα f. The operator Gνα is the Green operator for the boundary condition Lν . Then it follows that a function u is a solution of the problem (α − W )u = f in D, (∗∗) ′ ′ Lu = m(x ) Lν u + γ(x ) u = 0 on ∂D if and only if the function w =u−v

is a solution of the problem (α − W )w = 0 Lw = −Lv = −γ(x′ ) v

in D, on ∂D.

Thus, as in the proof of Theorem 1 we can reduce the study of problem (∗∗) to that of the equation LHα ψ = −LGνα f = −γ(x′ ) Gνα f

on ∂D.

By using the H¨older space theory of pseudo-differential operators as in the proof of Theorem 1, we can show that if condition (A) is satisfied, then the operator LHα is bijective in the framework of H¨older spaces. Therefore, we find that a unique solution u of problem (∗∗) can be expressed as follows: u = Gνα f − Hα LHα−1 (LGνα f ) .

This formula allows us to verify all the conditions of the generation theorems of Feller semigroups, especially the density of the domain D(W) in C0 (D \ M ). It is worth while pointing out that if we use instead of Gνα the Green operator 0 Gα for the Dirichlet problem as in the proof of Theorem 1, our proof would break down. We do not prove Theorems 3 and 4, since their proofs are essentially the same as those of Theorems 1 and 2, respectively. The content of this paper may be summarized in the following diagram:

Probability

Functional Analysis

Partial Differentital Equations

Markov process (Xt )

FellerR semigroup Tt f (·) = D pt (·, dy)f (y)

Infinitesimal generator Tt = exp[tA]

Semigroup property

Waldenfels operator Ventcel’ boundary condition

pt (x, dy) Markov transition function Markov property

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KAZUAKI TAIRA

1. Theory of Feller Semigroups This section provides a brief description of basic definitions and results about a class of semigroups (Feller semigroups) associated with Markov processes, which forms a functional analytic background for the proof of Theorems 1 and 2. The results discussed here are adapted from Taira [Ta1, Chapter 9]. 1.1 Markov Processes. In 1828 the English botanist R. Brown observed that pollen grains suspended in water move chaotically, incessantly changing their direction of motion. The physical explanation of this phenomenon is that a single grain suffers innumerable collisions with the randomly moving molecules of the surrounding water. A mathematical theory for Brownian motion was put forward by A. Einstein in 1905. Let p(t, x, y) be the probability density function that a one-dimensional Brownian particle starting at position x will be found at position y at time t. Einstein derived the following formula from statistical mechanical considerations: (y − x)2 1 exp − . p(t, x, y) = √ 2Dt 2πDt Here D is a positive constant determined by the radius of the particle, the interaction of the particle with surrounding molecules, temperature and the Boltzmann constant. This gives an accurate method of measuring Avogadro’s number by observing particles. Einstein’s theory was experimentally tested by J. Perrin between 1906 and 1909. Brownian motion was put on a firm mathematical foundation for the first time by N. Wiener in 1923. Let Ω be the space of continuous functions ω : [0, ∞) 7→ R with coordinates xt (ω) = ω(t) and let F be the smallest σ-algebra in Ω which contains all sets of the form {ω ∈ Ω : a ≤ xt (ω) < b}, t ≥ 0, a < b. Wiener constructed probability measures Px , x ∈ R, on F for which the following formula holds: Px {ω ∈ Ω : a1 ≤ xt1 (ω) < b1 , a2 ≤ xt2 (ω) < b2 , . . . , an ≤ xtn (ω) < bn } Z bn Z b1 Z b2 p(t1 , x, y1 )p(t2 − t1 , y1 , y2 ) . . . ... = a1

a2

an

p(tn − tn−1 , yn−1 , yn ) dy1 dy2 . . . dyn , 0 < t1 < t2 < . . . < tn < ∞.

This formula expresses the “starting afresh” property of Brownian motion that if a Brownian particle reaches a position, then it behaves subsequently as though that position had been its initial position. The measure Px is called the Wiener measure starting at x. Markov processes are an abstraction of the idea of Brownian motion. Let K be a locally compact, separable metric space and B the σ-algebra of all Borel sets in K, that is, the smallest σ-algebra containing all open sets in K. Let (Ω, F , P ) be a probability space. A function X defined on Ω taking values in K is called a random variable if it satisfies the condition {X ∈ E} = X −1 (E) ∈ F

for all E ∈ B.

FELLER SEMIGROUPS AND MARKOV PROCESSES

17

We express this by saying that X is F /B-measurable. A family {xt }t≥0 of random variables is called a stochastic process, and it may be thought of as the motion in time of a physical particle. The space K is called the state space and Ω the sample space. For a fixed ω ∈ Ω, the function xt (ω), t ≥ 0, defines in the state space K a trajectory or path of the process corresponding to the sample point ω. In this generality the notion of a stochastic process is of course not so interesting. The most important class of stochastic processes is the class of Markov processes which is characterized by the Markov property. Intuitively, this is the principle of the lack of any “memory” in the system. More precisely, (temporally homogeneous) Markov property is that the prediction of subsequent motion of a particle, knowing its position at time t, depends neither on the value of t nor on what has been observed during the time interval [0, t); that is, a particle “starts afresh”. Now we introduce a class of Markov processes which we will deal with in this book. Assume that we are given the following: (1) A locally compact, separable metric space K and the σ-algebra B of all Borel sets in K. A point ∂ is adjoined to K as the point at infinity if K is not compact, and as an isolated point if K is compact. We let K∂ = K ∪ {∂}, B∂ = the σ-algebra in K∂ generated by B. (2) The space Ω of all mappings ω : [0, ∞] → K∂ such that ω(∞) = ∂ and that if ω(t) = ∂ then ω(s) = ∂ for all s ≥ t. Let ω∂ be the constant map ω∂ (t) = ∂ for all t ∈ [0, ∞]. (3) For each t ∈ [0, ∞], the coordinate map xt defined by xt (ω) = ω(t), ω ∈ Ω. (4) For each t ∈ [0, ∞], a mapping ϕt : Ω → Ω defined by ϕt ω(s) = ω(t + s), ω ∈ Ω. Note that ϕ∞ ω = ω∂ and xt ◦ ϕs = xt+s for all t, s ∈ [0, ∞]. (5) A σ-algebra F in Ω and an increasing family {Ft }0≤t≤∞ of sub-σ-algebras of F . (6) For each x ∈ K∂ , a probability measure Px on (Ω, F ). We say that these elements define a (temporally homogeneous) Markov process X = (xt , F , Ft, Px ) if the following conditions are satisfied: (i) For each 0 ≤ t < ∞, the function xt is Ft /B∂ - measurable, that is, {xt ∈ E} ∈ Ft

for all E ∈ B∂ .

(ii) For each 0 ≤ t < ∞ and E ∈ B, the function (1.1)

pt (x, E) = Px {xt ∈ E}

is a Borel measurable function of x ∈ K. (iii) Px {ω ∈ Ω : x0 (ω) = x} = 1 for each x ∈ K∂ . (iv) For all t, h ∈ [0, ∞], x ∈ K∂ and E ∈ B∂ , we have Px {xt+h ∈ E | Ft } = ph (xt , E) a. e.,

18

KAZUAKI TAIRA

or equivalently Px (A ∩ {xt+h ∈ E}) =

Z

ph (xt (ω), E) dPx(ω),

A

A ∈ Ft .

Here is an intuitive way of thinking about the above definition of a Markov process. The sub-σ-algebra Ft may be interpreted as the collection of events which are observed during the time interval [0, t]. The value Px (A), A ∈ F , may be interpreted as the probability of the event A under the condition that a particle starts at position x; hence the value pt (x, E) expresses the transition probability that a particle starting at position x will be found in the set E at time t (see Figure 1.1 below). The function pt is called the transition function of the process X . The transition function pt specifies the probability structure of the process. The intuitive meaning of the crucial condition (iv) is that the future behavior of a particle, knowing its history up to time t, is the same as the behavior of a particle starting at xt (ω), that is, a particle starts afresh. A particle moves in the space K until it “dies” at the time when it reaches the point ∂; hence the point ∂ is called the terminal point. With this interpretation in mind, we let ζ(ω) = inf{t ∈ [0, ∞] : xt (ω) = ∂}. The random variable ζ is called the lifetime of the process X . E

........................................ ...... ....... ... ...... .......... . . . . . .................................... ...

..... ... ... ........ ... ... ........ ... ... .......... ... ....... ... ....... ... ....... ..... ....... .. ... ......... . ... ... .................. . ................ ... ................... ... ... ... .... ... .. ....... . . . . . . . . ...... .. . . . . . . . . . . . . . . . ...........

t

• x

Figure 1.1

1.2 Markov Transition Functions and Feller Semigroups. From the point of view of analysis, the transition function is something more convenient than the Markov process itself. In fact, it can be shown that the transition functions of Markov processes generate solutions of certain parabolic partial differential equations such as the classical diffusion equation; and, conversely, these differential equations can be used to construct and study the transition functions and the Markov processes themselves. Let (K, ρ) be a locally compact, separable metric space and B the σ-algebra of all Borel sets in K.

FELLER SEMIGROUPS AND MARKOV PROCESSES

19

A function pt (x, E), defined for all t ≥ 0, x ∈ K and E ∈ B, is called a (temporally homogeneous) Markov transition function on K if it satisfies the following four conditions: (a) pt (x, ·) is a non-negative measure on B and pt (x, K) ≤ 1 for each t ≥ 0 and each x ∈ K. (b) pt (·, E) is a Borel measurable function for each t ≥ 0 and each E ∈ B. (c) p0 (x, {x}) = 1 for each x ∈ K. (d) (The Chapman-Kolmogorov equation) For any t, s ≥ 0, any x ∈ K and any E ∈ B, we have Z (1.2) pt+s (x, E) = pt (x, dy)ps (y, E). K

Here is an intuitive way of thinking about the above definition of a Markov transition function. The value pt (x, E) expresses the transition probability that a physical particle starting at position x will be found in the set E at time t (see Figure 1.1). Equation (1.2) expresses the idea that a transition from the position x to the set E in time t + s is composed of a transition from x to some position y in time t, followed by a transition from y to the set E in the remaining time s; the latter transition has probability ps (y, E) which depends only on y (see Figure 1.2). Thus a particle “starts afresh”; this property is called the Markov property.

t+s

t

0

. ..... ......... ... ... .. ............................................................................................................................................................................................................................................... ....... ..... .. ... ...... ... ....... .... ..... ........ .. ...... .... ... .... ...... ...... ... .... .... ...... . . . . ... . . . . . . . . . .. . ..... .... .. .... ..... ... .. .... ... ..... . . . . . . . . . ... . . ..... ... . .. . . . . .. ......... . .... .. ... ... .... .. .. .... ... .... . . . . ... .... . . ... .. ... ... .... ..... ... .... ..... .... .. ... .... . .............................................................................................................................................................................................................. .. . ... . . ...... ... ..... .. ..... ... ...... .. . ... . . .. ... ... .. . ... ... .. .... .. ... .. .. ..... ..... ... . ... . .... . ... . ... . ... .... .... .. .. .. ..... ..... ... .. ... .. ..... ..... .... ... .... ..... .... ...... . . . . . . .... . . . . .. ..... .. ... .. . . . . . . . ..... ... .. . . ... ..... ..... .... .... ... ...... ..... . . ..... ... . . . . . . . . .. ..... .... .... .... .... .. ........... ...... .................. . . . . . ........... . ... . . .. ........ ........... ..................................................................................................................................................................................................................................................... . ... . . ... .. ... ... ... ... .... .... . .... . .... .... .... .... ..... .... ...... ..... . . . . . ......... .. ..... ............

..........E .....................

•y

• x

K

Figure 1.2 The Chapman–Kolmogorov equation (1.2) tells us that pt (x, K) is monotonically increasing as t ↓ 0, so that the limit p+0 (x, K) = limt↓0 pt (x, K) exists. A transition function pt is said to be normal if it satisfies the condition p+0 (x, K) = 1 for all x ∈ K. It is known that, for every Markov process, the function pt , defined by formula (1.1), is a transition function. Conversely, every normal transition function corresponds to some Markov process.

20

KAZUAKI TAIRA

We add a point ∂ to K as the point at infinity if K is not compact, and as an isolated point if K is compact; so the space K∂ = K ∪ {∂} is compact. Let C(K) be the space of real-valued, bounded continuous functions on K. The space C(K) is a Banach space with the supremum norm kf k∞ = sup |f (x)|. x∈K

We say that a function f ∈ C(K) converges to 0 as x → ∂ if, for each ε > 0, there exists a compact subset E of K such that |f (x)| < ε,

x ∈ K \ E,

and we then write limx→∂ f (x) = 0. We let C0 (K) = f ∈ C(K) : lim f (x) = 0 . x→∂

The space C0 (K) is a closed subspace of C(K); hence it is a Banach space. Note that C0 (K) may be identified with C(K) if K is compact. If we introduce a useful convention Any real-valued function f on K is extended to the space K∂ = K ∪ {∂} by setting f (∂) = 0,

then the space C0 (K) may be identified with the subspace of C(K∂ ) which consists of all functions f satisfying f (∂) = 0: C0 (K) = {f ∈ C(K∂ ) : f (∂) = 0} . Moreover, we can extend a Markov transition function pt on K to a Markov transition function p′t on K∂ as follows: ′ pt (x, E) = pt (x, E), x ∈ K, E ∈ B; p′t (x, {∂}) = 1 − pt (x, K), x ∈ K; ′ pt (∂, K) = 0, p′t (∂, {∂}) = 1.

Intuitively, this means that a Markovian particle moves in the space K until it “dies” at the time when it reaches the point ∂; hence the point ∂ is called the terminal point. Now we introduce some conditions on the measures pt (x, ·) related to continuity in x ∈ K, for fixed t ≥ 0. A Markov transition function pt is called a Feller function if the function Z Tt f (x) = pt (x, dy)f (y) K

is a continuous function of x ∈ K whenever f is in C(K), that is, if we have f ∈ C(K) =⇒ Tt f ∈ C(K).

FELLER SEMIGROUPS AND MARKOV PROCESSES

21

In other words, the Feller property is equivalent to saying that the measures pt (x, ·) depend continuously on x ∈ K in the usual weak topology, for every fixed t ≥ 0. We say that pt is a C0 -function if the space C0 (K) is an invariant subspace of C(K) for the operators Tt : f ∈ C0 (K) =⇒ Tt f ∈ C0 (K). The Feller or C0 -property deals with continuity of a Markov transition function pt (x, E) in x, and does not, by itself, have no concern with continuity in t. We give a necessary and sufficient condition on pt (x, E) in order that its associated operators {Tt }t≥0 be strongly continuous in t on the space C0 (K): lim kTt+s f − Tt f k∞ = 0, s↓0

f ∈ C0 (K).

A Markov transition function pt on K is said to be uniformly stochastically continuous on K if the following condition is satisfied: For each ε > 0 and each compact E ⊂ K, we have lim sup [1 − pt (x, Uε (x))] = 0, t↓0 x∈E

where Uε (x) = {y ∈ K : ρ(x, y) < ε} is an ε-neighborhood of x. Then we have the following (see [Ta1, Theorem 9.2.3]): Theorem 1.1. Let pt be a C0 -transition function on K. Then the associated operators {Tt }t≥0 , defined by the formula Z (1.3) Tt f (x) = pt (x, dy)f (y), f ∈ C0 (K), K

are strongly continuous in t on C0 (K) if and only if pt is uniformly stochastically continuous on K and satisfies the following condition: (L) For each s > 0 and each compact E ⊂ K, we have lim sup pt (x, E) = 0.

x→∂ 0≤t≤s

A family {Tt }t≥0 of bounded linear operators acting on C0 (K) is called a Feller semigroup on K if it satisfies the following three conditions: (i) Tt+s = Tt · Ts , t, s ≥ 0; T0 = I. (ii) The family {Tt } is strongly continuous in t for t ≥ 0: lim kTt+s f − Tt f k∞ = 0, s↓0

f ∈ C0 (K).

(iii) The family {Tt } is non-negative and contractive on C0 (K): f ∈ C0 (K), 0 ≤ f ≤ 1 on K =⇒ 0 ≤ Tt f ≤ 1 on K. The next theorem gives a characterization of Feller semigroups in terms of Markov transition functions (see [Ta1, Theorem 9.2.6]):

22

KAZUAKI TAIRA

Theorem 1.2. If pt is a uniformly stochastically continuous C0 -transition function on K and satisfies condition (L), then its associated operators {Tt }t≥0 form a Feller semigroup on K. Conversely, if {Tt }t≥0 is a Feller semigroup on K, then there exists a uniformly stochastically continuous C0 -transition pt on K, satisfying condition (L), such that formula (1.3) holds. 1.3 Generation Theorems of Feller Semigroups. If {Tt }t≥0 is a Feller semigroup on K, we define its infinitesimal generator A by the formula (1.4)

Au = lim t↓0

Tt u − u , t

provided that the limit (1.4) exists in the space C0 (K). More precisely, the generator A is a linear operator from the space C0 (K) into itself defined as follows. (1) The domain D(A) of A is the set D(A) = {u ∈ C0 (K) : the limit (1.4) exists} . Tt u − u , u ∈ D(A). t The next theorem is a version of the Hille–Yosida theorem adapted to the present context (see [Ta1, Theorem 9.3.1 and Corollary 9.3.2]): (2) Au = limt↓0

Theorem 1.3. (i) Let {Tt }t≥0 be a Feller semigroup on K and A its infinitesimal generator. Then we have the following: (a) The domain D(A) is everywhere dense in the space C0 (K). (b) For each α > 0, the equation (αI − A)u = f has a unique solution u in D(A) for any f ∈ C0 (K). Hence, for each α > 0, the Green operator (αI − A)−1 : C0 (K) −→ C0 (K) can be defined by the formula u = (αI − A)−1 f,

f ∈ C0 (K).

(c) For each α > 0, the operator (αI − A)−1 is non-negative on the space C0 (K): f ∈ C0 (K), f ≥ 0

on K =⇒ (αI − A)−1 f ≥ 0

on K.

(d) For each α > 0, the operator (αI − A)−1 is bounded on the space C0 (K) with norm 1 k(αI − A)−1 k ≤ . α (ii) Conversely, if A is a linear operator from the space C0 (K) into itself satisfying condition (a) and if there exists a constant α0 ≥ 0 such that, for all α > α0 , conditions (b) through (d) are satisfied, then the operator A is the infinitesimal generator of some Feller semigroup {Tt }t≥0 on K.

FELLER SEMIGROUPS AND MARKOV PROCESSES

23

Corollary 1.4. Let K be a compact metric space and let A be the infinitesimal generator of a Feller semigroup on K. Assume that the constant function 1 belongs to the domain D(A) of A and that we have, for some constant c, A 1 ≤ −c

on K.

Then the operator A′ = A + cI is the infinitesimal generator of some Feller semigroup on K. Finally, we give useful criteria in terms of the maximum principle (see [Ta1, Theorem 9.3.3 and Corollary 9.3.4]): Theorem 1.5. Let K be a compact metric space. Then we have the following assertions: (i) Let B be a linear operator from the space C(K) = C0 (K) into itself, and assume that (α) The domain D(B) of B is everywhere dense in the space C(K). (β) There exists an open and dense subset K0 of K such that if u ∈ D(B) takes a positive maximum at a point x0 of K0 , then we have Bu(x0 ) ≤ 0. Then the operator B is closable in the space C(K). (ii) Let B be as in part (i), and further assume that (β ′ ) If u ∈ D(B) takes a positive maximum at a point x′ of K, then we have Bu(x′ ) ≤ 0. (γ) For some α0 ≥ 0, the range R(α0 I − B) of α0 I − B is everywhere dense in the space C(K). Then the minimal closed extension B of B is the infinitesimal generator of some Feller semigroup on K. Corollary 1.6. Let A be the infinitesimal generator of a Feller semigroup on a compact metric space K and C a bounded linear operator on C(K) into itself. Assume that either C or A′ = A + C satisfies condition (β ′ ). Then the operator A′ is the infinitesimal generator of some Feller semigroup on K. 2. Theory of Pseudo-Differential Operators In this section we present a brief description of basic concepts and results of the H¨older space theory of pseudo-differential operators which will be used in the subsequent sections. In particular we introduce the notion of transmission property due to Boutet de Monvel [Bo], which is a condition about symbols in the normal direction at the boundary. Furthermore, we prove an existence and uniqueness theorem for a class of pseudo-differential operators which enters naturally in the construction of Feller semigroups. For detailed studies of pseudo-differential operators, the reader is referred to Chazarain–Piriou [CP], H¨ormander [Ho], Kumano-go [Ku], Rempel–Schulze [RS] and Taylor [Ty].

24

KAZUAKI TAIRA

2.1 Function Spaces. Let Ω be an open subset of Rn . A Lebesgue measurable function u on Ω is said to be essentially bounded if there exists a constant C > 0 such that |u(x)| ≤ C almost everywhere (a.e.) in Ω. We define ess supx∈Ω |u(x)| = inf{C : |u(x)| ≤ C a.e. in Ω}. We let L∞ (Ω) = the space of equivalence classes of essentially bounded, Lebesgue measurable functions on Ω. The space L∞ (Ω) is a Banach space with the norm kuk∞ = ess supx∈Ω |u(x)|. If m is a non-negative integer, we let W m,∞ (Ω) = the space of equivalence classes of functions u ∈ L∞ (Ω) all of whose derivatives ∂ α u,

|α| ≤ m, in the sense of distributions are

in L∞ (Ω).

The space W m,∞ (Ω) is a Banach space with the norm kukm,∞ =

X

|α|≤m

k∂ α uk∞ .

Here and in the following we use the shorthand ∂ , 1 ≤ j ≤ n, ∂xj ∂ α = ∂1α1 ∂2α2 . . . ∂nαn , α = (α1 , α2 , . . . , αn ), ∂j =

for derivatives on Rn . We remark that W 0,∞ (Ω) = L∞ (Ω);

k · k0,∞ = k · k∞ .

Now we let C(Ω) = the space of continuous functions on Ω. If k is a positive integer, we let C k (Ω) = the space of functions of class C k on Ω.

FELLER SEMIGROUPS AND MARKOV PROCESSES

25

Further we let C(Ω) = the space of functions in C(Ω) having continuous extensions to the closure Ω of Ω. If k is a positive integer, we let C k (Ω) = the space of functions in C k (Ω) all of whose derivatives of order ≤ k have continuous extensions to Ω. If Ω is bounded, then the space C k (Ω) is a Banach space with the norm kukC k (Ω) = sup |∂ α u(x)|. x∈Ω |α|≤k

Let 0 < θ < 1. A function u defined on Ω is said to be H¨ older continuous with exponent θ if the quantity [u]θ;Ω = sup x,y∈Ω x6=y

|u(x) − u(y)| |x − y|θ

is finite. We say that u is locally H¨ older continuous with exponent θ if it is H¨older continuous with exponent θ on compact subsets of Ω. We let C θ (Ω) = the space of functions in C(Ω) which are locally H¨older continuous with exponent θ on Ω. If k is a positive integer, we let C k+θ (Ω) = the space of functions in C k (Ω) all of whose k-th order derivatives are locally H¨older continuous with exponent θ on Ω. Further we let C θ (Ω) = the space of functions in C(Ω) which are H¨older continuous with exponent θ on Ω. If k is a positive integer, we let C k+θ (Ω) = the space of functions in C k (Ω) all of whose k-th order derivatives are H¨older continuous with exponent θ on Ω.

26

KAZUAKI TAIRA

If Ω is bounded, then the space C k+θ (Ω) is a Banach space with the norm kukC k+θ (Ω) = kukC k (Ω) + sup [∂ α u]θ;Ω . |α|=k

If M is an n-dimensional compact smooth manifold without boundary and m is a non-negative integer, then the spaces W m,∞ (M ) and C m+θ (M ) are defined respectively to be locally the spaces W m,∞ (Rn ) and C m+θ (Rn ), upon using local coordinate systems flattening out M , together with a partition of unity. The norms of the spaces W m,∞ (M ) and C m+θ (M ) will be denoted by k·km,∞ and k·kC m+θ (M ) , respectively. We recall the following results (see [Tr]): (I) If k is a positive integer, then we have W k,∞ (M ) = {ϕ ∈ C k−1 (M ) : max

|α|≤k−1

sup x,y∈M x6=y

|∂ α ϕ(x) − ∂ α ϕ(y)| < ∞}, |x − y|

where |x − y| is the geodesic distance between x and y with respect to the Riemannian metric of M . (II) The space C k+θ (M ) is a real interpolation space between W k,∞ (M ) and W k+1,∞ (M ); more precisely, we have C k+θ (M ) = W k,∞ (M ), W k+1,∞(M ) θ,∞ K(t, u) k,∞ = u∈W (M ) : sup 0 where K(t, u) =

inf

u=u0 +u1

(ku0 kk,∞ + tku1 kk+1,∞ ) .

2.2 Pseudo-Differential Operators. Let Ω be an open subset of Rn . If m ∈ R and 0 ≤ δ < ρ ≤ 1, we let m Sρ,δ (Ω × RN ) = the set of all functions a ∈ C ∞ (Ω × RN ) with the property

that, for any compact K ⊂ Ω and multi-indices α, β, there exists a constant CK,α,β > 0 such that we have, for all x ∈ K and θ ∈ RN , α β ∂θ ∂x a(x, θ) ≤ CK,α,β (1 + |θ|)m−ρ|α|+δ|β| .

m The elements of Sρ,δ (Ω × RN ) are called symbols of order m. We set \ m S −∞ (Ω × RN ) = Sρ,δ (Ω × RN ). m∈R

For example, if ϕ(ξ) ∈ S(RN ), then it follows that ϕ(ξ) ∈ S −∞ (Ω × RN ). More precisely, we have the following: S −∞ (Ω × RN ) = C ∞ (Ω, S(RN )).

FELLER SEMIGROUPS AND MARKOV PROCESSES

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m

If aj ∈ Sρ,δj (Ω × RN ) is a sequence of symbols of decreasing order, then there m0 exists a symbol a ∈ Sρ,δ (Ω × RN ), unique modulo S −∞ (Ω × RN ), such that we have, for all k > 0, k−1 X mk a− aj ∈ Sρ,δ (Ω × RN ). j=0

In this case, we write

∞ X

a∼

aj .

j=0

P The formal sum j aj is called an asymptotic expansion of a. m A symbol a(x, θ) ∈ S1,0 (Ω × RN ) is said to be classical if there exist smooth functions aj (x, θ), positively homogeneous of degree m − j in θ for |θ| ≥ 1, such that ∞ X aj . a∼ j=0

The homogeneous function a0 of degree m is called the principal part of a. We let m Scl (Ω × RN ) = the set of all classical symbols of order m. m m Moreover, if Scl (RN ) is the subspace of all x-independent elements of Scl (Ω×RN ), then we have the following: m m (RN )). (Ω × RN ) = C ∞ (Ω, Scl Scl

Let Ω be an open subset of Rn and m ∈ R. A pseudo-differential operator of order m on Ω is a Fourier integral operator of the form ZZ Au(x) = ei(x−y)·ξ a(x, y, ξ)u(y) dydξ , u ∈ C0∞ (Ω), Ω×Rn

m (Ω×Ω×Rn ). Here the integral is taken in the sense of oscillatory with some a ∈ Sρ,δ integrals. We let

Lm ρ,δ (Ω) = the set of all pseudo-differential operators of order m on Ω, and set L−∞ (Ω) =

\

Lm ρ,δ (Ω).

m∈R

Recall that a continuous linear operator A : C0∞ (Ω) −→ D ′ (Ω) is said to be properly supported if the following two conditions are satisfied: (a) For any compact subset K of Ω, there exists a compact subset K ′ of Ω such that supp v ⊂ K =⇒ supp Av ⊂ K ′ .

28

KAZUAKI TAIRA

(b) For any compact subset K ′ of Ω, there exists a compact subset K ⊃ K ′ of Ω such that supp v ∩ K = ∅ =⇒ supp Av ∩ K ′ = ∅. m If A ∈ Lm ρ,δ (Ω), we can choose a properly supported operator A0 ∈ Lρ,δ (Ω) such that A − A0 ∈ L−∞ (Ω), and define

σ(A) = the equivalence class of the complete symbol of A0 m in the factor class Sρ,δ (Ω × Rn )/S −∞ (Ω × Rn ).

The equivalence class σ(A) does not depend on the operator A0 chosen, and is called the complete symbol of A. We shall often identify the complete symbol σ(A) with a representative in the m class Sρ,δ (Ω × Rn ) for notational convenience, and call any member of σ(A) a complete symbol of A. A pseudo-differential operator A ∈ Lm 1,0 (Ω) is said to be classical if its complete m (Ω × Rn ). symbol σ(A) has a representative in the class Scl We let Lm cl (Ω) = the set of all classical pseudo-differential operators of order m on Ω. If A ∈ Lm cl (Ω), then the principal part of σ(A) has a canonical representative σA (x, ξ) ∈ C ∞ (Ω × (Rn \ {0})) which is positively homogeneous of degree m in the variable ξ. The function σA (x, ξ) is called the homogeneous principal symbol of A. Now we define the concept of a pseudo-differential operator on a manifold, and transfer all the machinery of pseudo-differential operators to manifolds. Let M be an n-dimensional, compact smooth manifold without boundary and 1 − ρ ≤ δ < ρ ≤ 1. A continuous linear operator A : C ∞ (M ) −→ C ∞ (M ) is called a pseudo-differential operator of order m ∈ R if it satisfies the following two conditions: (i) The distribution kernel of A is of class smooth off the diagonal ∆M = {(x, x) : x ∈ M } in M × M . (ii) For any chart (U, χ) on M , the mapping Aχ : C0∞ (χ(U )) −→ C ∞ (χ(U )) u 7−→ A(u ◦ χ) ◦ χ−1

belongs to the class Lm ρ,δ (χ(U )). C0∞ (U ) x ∗ χ

A

−−−−→

C ∞ (U ) χ∗ y

C0∞ (χ(U )) −−−−→ C ∞ (χ(U )) Aχ

FELLER SEMIGROUPS AND MARKOV PROCESSES

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Here χ∗ v = v ◦ χ is the pull-back of v by χ and χ∗ u = u ◦ χ−1 is the push-forward of u by χ, respectively. We let Lm ρ,δ (M ) = the set of all pseudo-differential operators of order m on M , and set L−∞ (M ) =

\

Lm ρ,δ (M ).

m∈R

Some results about pseudo-differential operators on Rn are also true for pseudodifferential operators on M , since pseudo-differential operators on M are defined to be locally pseudo-differential operators on Rn . For example we have the following results: (1) A pseudo-differential operator A extends to a continuous linear operator A : D ′ (M ) −→ D ′ (M ). (2) sing supp Au ⊂ sing supp u , u ∈ D ′ (M ). (3) A continuous linear operator A : C ∞ (M ) −→ D ′ (M ) is a regularizer if and only if it is in L−∞ (M ). (4) The class Lm ρ,δ (M ), 1 − ρ ≤ δ < ρ ≤ 1, is stable under the operations of composition of operators and taking the transpose or adjoint of an operator. (5) A pseudo-differential operator A ∈ Lm 1,0 (M ) extends to a continuous linear k+θ k−m+θ operator A : C (M ) → C (M ) for any integer k ≥ m. A pseudo-differential operator A ∈ Lm 1,0 (M ) is said to be classical if, for any chart (U, χ) on M , the mapping Aχ : C0∞ (χ(U )) −→ C ∞ (χ(U )) belongs to the class Lm cl (χ(U )). We let Lm cl (M ) = the set of all classical pseudo-differential operators of order m on M . We observe that L−∞ (M ) =

\

Lm cl (M ).

m∈R

Let A ∈ Lm cl (M ). If (U, χ) is a chart on M , there is associated a homogeneous principal symbol σAχ ∈ C ∞ (χ(U ) × (Rn \ {0})). Then, by smoothly patching together the functions σAχ we can obtain a smooth function σA (x, ξ) on T ∗ (M ) \ {0} = {(x, ξ) ∈ T ∗ (M ) : ξ 6= 0}, which is positively homogeneous of degree m in the variable ξ. The function σA (x, ξ) is called the homogeneous principal symbol of A. A classical pseudo-differential operator A ∈ Lm cl (M ) is said to be elliptic of order m if its homogeneous principal symbol σA (x, ξ) does not vanish on the bundle T ∗ (M ) \ {0} of non-zero cotangent vectors. Then we have the following: (6) An operator A ∈ Lm cl (M ) is elliptic if and only if there exists a parametrix −m B ∈ Lcl (M ) for A: AB ≡ I mod L−∞ (M ), BA ≡ I

mod L−∞ (M ).

30

KAZUAKI TAIRA

Finally, we introduce the notion of transmission property, due to Boutet de Monvel [Bo], which is a condition about symbols in the normal direction at the boundary. We let m m (Rn+ × Rn+ × Rn ) S1,0 (Rn+ × Rn+ × Rn ) = the space of symbols in S1,0

m which have an extension in S1,0 (Rn × Rn × Rn ).

m We say that a symbol a(x, y, ξ) ∈ S1,0 (Rn+ ×Rn+ ×Rn ) has the transmission property with respect to the boundary Rn−1 if the function a(x, y, ξ) and all its derivatives admit an expansion of the form α β ∂ ∂ a(x, y, ξ) x=y ∂x ∂y xn =0

=

m X j=0

bj (x′ , ξ ′ )ξnj +

∞ X

k

ak (x′ , ξ ′ )

k=−∞

(< ξ ′ > −iξn )

(< ξ ′ > +iξn )

k+1

,

m−j where bj ∈ S1,0 (Rn−1 × Rn−1 ) and the ak form a rapidly decreasing sequence in m+1 S1,0 (Rn−1 × Rn−1 ) with respect to k, and < ξ ′ >= (1 + |ξ ′ |2 )1/2 . We let m n n Lm 1,0 (R+ ) = the space of pseudo-differential operators in L1,0 (R+ ) which can n be extended to a pseudo-differential operator in Lm 1,0 (R ). n A pseudo-differential operator A ∈ Lm 1,0 (R+ ) is said to have the transmission property with respect to the boundary Rn−1 if any complete symbol of A has the transmission property with respect to the boundary Rn−1 . It is known that a n pseudo-differential operator A ∈ Lm 1,0 (R+ ) has the transmission property with respect to the boundary Rn−1 if the restriction of A(u0 ) to Rn+ has a smooth extension to Rn for every u ∈ C0∞ (Rn+ ), where u0 is the extension of u to Rn by zero outside Rn+ .

...................................... . . . . . ..... ..... .... ..... ... .. ... M Ω .. ... .. ... .... .... .... .... .... .... .... . .. ... .... ... .. ........ .... .... . .... . .. .. .. . .... ..... ...... .. ∂Ω ............ . . . . ...... ... ........ . . . . . . ... ... ................ ............................... .... ... . ... .. ... . .... ... . ..... . . ... ....... .................................... Figure 2.1

FELLER SEMIGROUPS AND MARKOV PROCESSES

31

It should be emphasized that the notion of transmission property may be transferred to manifolds with boundary. Indeed, if Ω is a relatively compact open subset of an n-dimensional paracompact smooth manifold M without boundary (see Figure 2.1), then the notion of transmission property can be extended to the class Lm 1,0 (M ), upon using local coordinate systems flattening out the boundary ∂Ω. Then we have the following results (see [Bo], [RS]): (I) If a pseudo-differential operator A ∈ Lm 1,0 (M ) has the transmission property with respect to the boundary ∂Ω, then the operator AΩ : C ∞ (Ω) −→ C ∞ (Ω) u 7−→ A(u0 )|Ω

maps C ∞ (Ω) continuously into itself, where u0 is the extension of u to M by zero outside Ω. (II) If a pseudo-differential operator A ∈ Lm 1,0 (M ) has the transmission property, k+θ then the operator AΩ maps C (Ω) continuously into C k−m+θ (Ω) for any integer k ≥ m and 0 < θ < 1. 2.3 Unique Solvability Theorem for Pseudo-Differential Operators. The next result will play an essential role in the construction of Feller semigroups in Sections 3 and 4 (see [Ta3, Theorem 2.1]): Theorem 2.1. Let T be a classical pseudo-differential operator of second order on an n-dimensional, compact smooth manifold M without boundary such that T = P + S, where: (a) The operator P is a second-order degenerate elliptic differential operator on M with non-positive principal symbol, and P 1 ≤ 0 on M . (b) The operator S is a classical pseudo-differential operator of order 2 − κ, κ > 0, on M and its distribution kernel s(x, y) is non-negative off the diagonal ∆M = {(x, x) : x ∈ M } in M × M . (c) T 1 = P 1 + S1 ≤ 0 on M . Then, for each integer k ≥ 1, there exists a constant λ = λ(k) > 0 such that, for any f ∈ C k+θ (M ), we can find a function ϕ ∈ C k+θ (M ) satisfying the equation (T − λI)ϕ = f

on M ,

and the estimate kϕkC k+θ (M ) ≤ Ck+θ (λ)kf kC k+θ (M ) . Here Ck+θ (λ) > 0 is a constant independent of f . Proof. We prove Theorem 2.1 by using a method of elliptic regularization (see [OR, Chapter I]), just as in the proof of Th´eor`eme 4.5 of Cancelier [Ca]. So we only give a sketch of the proof. (1) We recall the following results:

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KAZUAKI TAIRA

Theorem 2.2. Let T = P + S be a classical pseudo-differential operator of second order on M as in Theorem 2.1. Assume that T 1 = P 1 + S1 < 0

on M .

Then we have, for all ϕ ∈ C 2 (M ), kϕkC(M ) ≤

kT ϕkC(M ) . − maxM T 1

Theorem 2.2 is a compact manifold version of Theorem A.2 in Appendix. Theorem 2.3. Let T = P + S be a classical pseudo-differential operator of second order on M as in Theorem 2.1. Assume that the operator T is elliptic on M and satisfies the condition T 1 = P 1 + S1 < 0 on M . Then, for each integer k ≥ 0, the operator T : C k+2+θ (M ) −→ C k+θ (M ) is bijective. Since T is elliptic and its principal symbol is real, it follows from an application of Taira [Ta1, Corollary 6.7.12] that ind T = dim N (T ) − codim R(T ) = 0. However, Theorem 2.2 tells us that T is injective, that is, dim N (T ) = 0. Hence we obtain that codim R(T ) = 0, which proves that T is surjective. (2) First, we prove Theorem 2.1 for the space W 1,∞ (M ): Claim I. There exists a constant λ = λ(1) > 0 such that for any f ∈ W 1,∞ (M ) we can find a function ϕ ∈ W 1,∞ (M ) satisfying (T − λI)ϕ = f

on M ,

and kϕk1,∞ ≤ C1 kf k1,∞ . Here C1 > 0 is a constant independent of f . Proof. (2-i) Let {(Uα , χα )}ℓα=1 be a finite open covering of M by local charts, and let {σα }ℓα=1 be a family of functions in C ∞ (M × M ) such that supp σα ⊂ Uα × Uα , and ℓ X

α=1

σα (x, y) = 1 in a neighborhood of the diagonal ∆M = {(x, x) : x ∈ M }.

FELLER SEMIGROUPS AND MARKOV PROCESSES

33

Then the operator T = P + S can be written, in terms of local coordinates (x1 , x2 , . . . , xn ), in the form n X ∂ϕ ∂ 2ϕ β i (x) (x) + (x) + γ(x)ϕ(x) α (x) T ϕ(x) = ∂xi ∂xj ∂xi i=1 i,j=1 Z n X ∂ϕ + s(x, y) ϕ(y) − σ(x, y) ϕ(x) + (yi − xi ) (x) dy. ∂x i M i=1 n X

ij

Here: ij (a) The α are the components of a smooth symmetric contravariant tensor of 2 type 0 on M satisfying n X

i,j=1

ij

α (x)ξi ξj ≥ 0,

x ∈ M, ξ =

n X j=1

ξj dxj ∈ Tx∗ (M ),

where Tx∗ (M ) is the cotangent space of M at x. Pℓ (b) σ(x, y) = α=1 σα (x, y). (c) The density dy isR a strictly positive density on M . (d) T 1(x) = γ(x) + M s(x, y)[1 − σ(x, y)]dy ≤ 0 on M . Furthermore, it should be noticed that there exists a constant C > 0 such that the distribution kernel s(x, y) of S ∈ L2−κ cl (M ), κ > 0, satisfies the estimate C , |x − y|n+2−κ

|s(x, y)| ≤

(x, y) ∈ (M × M ) \ ∆M ,

where |x − y| is the geodesic distance between x and y with respect to the Riemannian metric of M (see [CM, Chapitre IV, Proposition 1]). Hence we find that the integral Z

M

n X ∂ϕ (x) dy s(x, y) ϕ(y) − σ(x, y) ϕ(x) + (yi − xi ) ∂x i i=1

is absolutely convergent, since κ > 0 and σ(x, y) = 1 in a neighborhood of the diagonal ∆M . Now, for each ϕ ∈ C 1 (M ) we introduce a continuous function B(ϕ, ϕ)(x) = 2

n X

αij (x)

i,j=1

+

Z

M

∂ϕ ∂ϕ (x) (x) ∂xi ∂xj 2

s(x, y) (ϕ(y) − ϕ(x)) dy − T 1(x) · ϕ(x)2 ,

x ∈ M.

It should be noticed that the function B(ϕ, ϕ) is non-negative on M for all ϕ ∈ C 1 (M ). The next result may be proved just as in the proof of Th´eor`eme 4.1 of Cancelier [Ca].

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KAZUAKI TAIRA

Lemma 2.4. Let {Xj }rj=1 be a family of real smooth vector fields on M such that the Xj span the tangent space Tx (M ) at each point x of M . If ϕ ∈ C ∞ (M ), we let p1 (x) =

r X j=1

and R1 (x) = T p1 (x) −

|Xj ϕ(x)|2 ,

r X

x ∈ M,

B(Xj ϕ, Xj ϕ)(x),

j=1

x ∈ M.

Then, for each η > 0, there exist constants β0 > 0 and β1 > 0 such that we have, for all ϕ ∈ C ∞ (M ), (2.1)

|R1 (x)| ≤ η

r X j=1

B(Xj ϕ, Xj ϕ)(x) + β0 kϕk2C(M )

1 + β1 kϕk2C 1 (M ) + kT ϕk2C 1 (M ) , 2

x ∈ M.

Remark 2.5. The constants β0 and β1 are uniform for the operators T + εΛ, 0 ≤ ε ≤ 1, where Λ is a second-order elliptic differential operator on M defined by the formula Λ=− =

r X

Xj∗ Xj

j=1 r X

Xj2

j=1

+

r X j=1

div Xj · Xj .

(2-ii) First, let f be an arbitrary element of C ∞ (M ). Since the operator T + εΛ − λI is elliptic for all ε > 0 and (T + εΛ − λI)1 = T 1 − λ < 0 on M for λ > 0, it follows from an application of Theorem 2.3 that we can find a unique function ϕε ∈ C ∞ (M ) such that (T + εΛ − λI)ϕε = f

on M .

Furthermore, by applying Theorem 2.2 to the operator T + εΛ − λI we obtain that (2.2)

kϕε kC(M ) ≤

1 kf kC(M ) , λ

since (T + εΛ − λI)1 = T 1 − λ ≤ −λ on M . Let x0 be a point of M at which the function p1 (x) =

r X j=1

|Xj ϕ(x)|2

FELLER SEMIGROUPS AND MARKOV PROCESSES

35

attains its positive maximum. Then we have Λp1 (x0 ) =

X r j=1

Xj2

p1 (x0 ) ≤ 0

and also n X

T p1 (x0 ) =

αij (x0 )

i,j=1

∂ 2 p1 (x0 ) + γ(x0 )p1 (x0 ) ∂xi ∂xj

Z

s(x0 , y)[p1 (y) − σ(x0 , y)p1 (x0 )]dy Z ≤ γ(x0 ) + s(x0 , y)[1 − σ(x0 , y)]dy p1 (x0 ) M Z + s(x0 , y)[p1 (y) − p1 (x0 )]dy +

M

M

≤ T 1(x0 ) · p1 (x0 ).

Hence, by using inequality (2.1) with η := 1/2 (and Remark 2.5) and inequality (2.2) we obtain that λp1 (x0 ) ≤ (λ − T 1(x0 ))p1 (x0 ) − εΛp1 (x0 )

≤ (λ − T − εΛ)p1 (x0 ) r X B(Xj ϕε , Xj ϕε )(x0 ) = − (T + εΛ − λ)p1 (x0 ) − j=1

− ≤−

r X

B(Xj ϕε , Xj ϕε )(x0 )

j=1 r X

1 2

j=1

B(Xj ϕε , Xj ϕε )(x0 ) + β0 kϕε k2C(M )

1 + β1 kϕε k2C 1 (M ) + kf k2C 1 (M ) 2 1 β0 ≤ 2 kf k2C(M ) + β1 kϕε k2C 1 (M ) + kf k2C 1 (M ) . λ 2 This proves that (2.3)

(λ − β1 )kϕε k2C 1 (M ) ≤ λ kϕε k2C(M ) + p1 (x0 ) − β1 kϕε k2C 1 (M ) ≤

1 β0 1 kf k2C(M ) + 2 kf k2C(M ) + kf k2C 1 (M ) . λ λ 2

Therefore, if λ > 0 is so large that λ > β1 ,

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KAZUAKI TAIRA

then it follows from inequality (2.3) that (2.4)

kϕε k2C 1 (M ) ≤ Ckf k2C 1 (M ) ,

where C > 0 is a constant independent of ε > 0. (2-iii) Now let f be an arbitrary element of W 1,∞ (M ). Then we can find a ∞ sequence {fp }∞ p=1 in C (M ) such that fp −→ f in C(M ), kfp kC 1 (M ) ≤ kf k1,∞ . If ϕεp is a unique solution in C ∞ (M ) of the equation (2.5)

(T + εΛ − λI)ϕεp = fp

on M ,

it follows from an application of inequality (2.4) that kϕεp k2C 1 (M ) ≤ Ckfp k2C 1 (M ) ≤ Ckf k21,∞ . This proves that the sequence {ϕεp } is uniformly bounded and equicontinuous. Hence, by virtue of the Ascoli–Arzel`a theorem we can choose a subsequence {ϕε′ p′ } which converges uniformly to a function ϕ ∈ C(M ), as ε′ ↓ 0 and p′ → ∞. Furthermore, since the unit ball in L2 (M ) is sequentially weakly compact (see [Yo, Chapter V, Section 2, Theorem 1]), we may assume that the sequence {∂j ϕε′ p′ } converges weakly to a function ψj in L2 (M ), for each 1 ≤ j ≤ n. Then we have ∂j ϕ = ψj ∈ L2 (M ),

1 ≤ j ≤ n.

On the other hand it is easy to verify that the set √ K = {g ∈ L2 (M ) : kgk∞ ≤ C kf k1,∞ }

is convex and strongly closed in L2 (M ). Thus it follows from an application of Mazur’s theorem (see [Yo, Chapter V, Section 1, Theorem 11]) that the set K is weakly closed in L2 (M ). However, we have ∂j ϕε′ p′ ∈ K, ∂j ϕε′ p′ −→ ψj weakly in L2 (M ) for each 1 ≤ j ≤ n. Hence we find that that is,

∂j ϕ = ψj ∈ K, k∂j ϕk∞ ≤

√

1 ≤ j ≤ n,

C kf k1,∞ ,

Summing up, we have proved that ϕ ∈ W 1,∞ (M ),

1 ≤ j ≤ n.

kϕk1,∞ ≤ C1 kf k1,∞ , where C1 > 0 is a constant independent of f . Finally, by letting ε′ ↓ 0 and p′ → ∞ in the equation

(2.5′ )

we obtain that

(T + ε′ Λ − λI)ϕε′ p′ = fp′ (T − λI)ϕ = f

on M ,

on M .

The proof of Claim I is complete. (3) Similarly we can prove Theorem 2.1 for the spaces W m,∞ (M ) where m ≥ 2:

FELLER SEMIGROUPS AND MARKOV PROCESSES

37

Claim II. For each integer m ≥ 2, there exists a constant λ = λ(m) > 0 such that for any f ∈ W m,∞ (M ) we can find a function ϕ ∈ W m,∞ (M ) satisfying (T − λI)ϕ = f

and

on M ,

kϕkm,∞ ≤ Cm kf km,∞ .

Here Cm > 0 is a constant independent of f .

(4) Therefore, Theorem 2.1 follows from Claims I and II by a well-known interpolation argument, since the space C k+θ (M ) is a real interpolation space between the spaces W k,∞ (M ) and W k+1,∞ (M ): C k+θ (M ) = W k,∞ (M ), W k+1,∞(M ) θ,∞ . 2.4 Positive Borel Kernels and the Positive Maximum Principle. Let Ω be an open subset of Rn , and let Bloc (Ω) = the space of Borel-measurable functions in Ω which are bounded on compact subsets of Ω. Let B be the σ-algebra of all Borel sets in Ω. A positive Borel kernel on Ω is a mapping x 7−→ s(x, dy)

of Ω into the space of non-negative measures on B such that, for each X ∈ B, the function s(·, X) is Borel-measurable on Ω. Now we assume that a positive Borel kernel s(x, dy) satisfies the following two conditions: (NS1) s(x, {x}) = 0 for all x ∈ Ω. (NS2) For all non-negative functions f in C0 (Ω), the function Z x 7−→ s(x, dy)|y − x|2 f (y), x ∈ Ω, Ω

belongs to the space Bloc (Ω). Let σ(x, y) be a smooth function on Ω × Ω such that (a) 0 ≤ σ(x, y) ≤ 1 on Ω × Ω. (b) σ(x, y) = 1 in a neighborhood of the diagonal {(x, x) : x ∈ Ω} in Ω × Ω. (c) For any compact subset K of Ω, there exists a compact subset K ′ of Ω such that the functions σx (·) = σ(x, ·), x ∈ K, have their support in K ′ . Then, by using Taylor’s formula and condition (NS2) we can define a linear operator S : C02 (Ω) −→ Bloc (Ω) by the formula (2.6)

n X ∂u (x)(yi − xi ) Su(x) = s(x, dy) u(y) − σ(x, y) u(x) + ∂x i Ω i=1

Z

"

u ∈ C02 (Ω).

!#

,

The next three theorems give useful characterizations of linear operators A from C02 (Ω) into Bloc (Ω) which satisfy the positive maximum principle (cf. [BCP, Th´eor`eme I, Th´eor`eme II and Th´eor`eme III]):

38

KAZUAKI TAIRA

Theorem 2.6. Let A be a linear operator from C02 (Ω) into Bloc (Ω). Then the following two assertions are equivalent: (i) A : C02 (Ω) → Bloc (Ω) is continuous and satisfies the condition (2.7)

x0 ∈ Ω, u ∈ C02 (Ω), u ≥ 0 in Ω and x0 6∈ supp u =⇒ Au(x0 ) ≥ 0.

(ii) There exist a second-order differential operator P : C 2 (Ω) → Bloc (Ω) and a positive Borel kernel s(x, dy), having properties (NS1) and (NS2), such that the operator A is written in the form (2.8)

x ∈ Ω, u ∈ C02 (Ω).

Au(x) = P u(x) + Su(x),

Theorem 2.7. Let V be a linear subspace of C02 (Ω) which contains C0∞ (Ω). Assume that A is a linear operator from V into Bloc (Ω) and satisfies the condition (2.9)

x0 ∈ Ω, u ∈ V, u ≥ 0 in Ω and u(x0 ) = 0 =⇒ Au(x0 ) ≥ 0.

Then the operator A can be extended uniquely to a continuous linear operator A : C02 (Ω) −→ Bloc (Ω) which still satisfies condition (2.9) for all u ∈ C02 (Ω): (2.9′ )

x0 ∈ Ω, u ∈ C02 (Ω), u ≥ 0 in Ω and u(x0 ) = 0 =⇒ Au(x0 ) ≥ 0.

Theorem 2.8. Let A be a linear operator from C02 (Ω) into Bloc (Ω) of the form (2.8), where P : C 2 (Ω) → Bloc (Ω) is a second-order differential operator on Ω and the operator S is defined by formula (2.6), with a positive Borel kernel s(x, dy) having properties (NS1) and (NS2). Then we have the following assertions: (i) The operator A satisfies condition (2.7) if and only if the principal symbol of P is non-positive. (ii) The operator A satisfies the condition (PM)

x0 ∈ Ω, v ∈ C02 (Ω) and v(x0 ) = sup v ≥ 0 =⇒ Av(x0 ) ≤ 0 Ω

if and only if the principal symbol of P is non-positive and P satisfies the following two conditions: P 1(x) ≤ 0,

x ∈ Ω, Z A1 = P 1(x) + s(x, dy) [1 − σ(x, y)] ≤ 0, Ω

x ∈ Ω.

3. Proof of Theorem 1 In this section we prove Theorem 1. First, we reduce the problem of construction of Feller semigroups to the problem of unique solvability for the boundary value problem (α − W )u = f in D, (λ − L)u = ϕ

on ∂D,

and then prove existence theorems for Feller semigroups. Here α is a positive number and λ is a non-negative number.

FELLER SEMIGROUPS AND MARKOV PROCESSES

39

3.1 General Existence Theorem for Feller Semigroups. The purpose of this subsection is to give a general existence theorem for Feller semigroups (Theorem 3.14) in terms of boundary value problems, following Taira [Ta1, Section 9.6] (cf. [BCP], [SU]). First, we consider the following Dirichlet problem: For given functions f and ϕ defined in D and on ∂D, respectively, find a function u in D such that (D)

(α − W )u = f u=ϕ

in D, on ∂D,

where α > 0. The next theorem summarizes the basic facts about the Dirichlet problem in the framework of H¨ older spaces (see [BCP, Th´eor`eme XV]): Theorem 3.1. Let k be an arbitrary non-negative integer and 0 < θ < 1. For any f ∈ C k+θ (D) and any ϕ ∈ C k+2+θ (∂D), problem (D) has a unique solution u in C k+2+θ (D). Theorem 3.1 with k := 0 tells us that problem (D) has a unique solution u in (D) for any f ∈ C θ (D) and any ϕ ∈ C 2+θ (∂D), 0 < θ < 1. Therefore, we can C introduce linear operators 2+θ

G0α : C θ (D) −→ C 2+θ (D), and Hα : C 2+θ (∂D) −→ C 2+θ (D) as follows. (a) For any f ∈ C θ (D), the function G0α f ∈ C 2+θ (D) is the unique solution of the problem (3.1)

(α − W )G0α f = f G0α f = 0

in D, on ∂D.

(b) For any ϕ ∈ C 2+θ (∂D), the function Hα ϕ ∈ C 2+θ (D) is the unique solution of the problem (3.2)

(α − W )Hα ϕ = 0 in D, Hα ϕ = ϕ on ∂D.

The operator G0α is called the Green operator and the operator Hα is called the harmonic operator , respectively. Then we have the following results (cf. [Ta1, Lemmas 9.6.2 and 9.6.3]): Lemma 3.2. The operator G0α , α > 0, considered from C(D) into itself, is nonnegative and continuous with norm

0 0

G = G 1 = max G0 1(x) . α α α ∞ x∈D

40

KAZUAKI TAIRA

Proof. Let f be an arbitrary function in C θ (D) such that f ≥ 0 on D. Then, by applying Theorem A.1 (the weak maximum principle) with W := W − α to the function −G0α f we obtain from formula (3.1) that G0α f ≥ 0 on D. This proves the non-negativity of G0α . Since G0α is non-negative, we have, for all f ∈ C θ (D), −G0α kf k∞ ≤ G0α f ≤ G0α kf k∞

on D.

This implies the continuity of G0α with norm kG0α k = kG0α 1k∞ . The proof is complete.

Lemma 3.3. The operator Hα , α > 0, considered from C(∂D) into C(D), is non-negative and continuous with norm kHα k = kHα 1k∞ = max Hα 1(x) . x∈D

More precisely, we have the following (see [BCP, Proposition III.1.6]): Theorem 3.4. (i) (a) The operator G0α , α > 0, can be uniquely extended to a non-negative, bounded linear operator on C(D) into itself, denoted again by G0α , with norm (3.3)

0 0

Gα = Gα 1 ≤ 1 . ∞ α

(b) For any f ∈ C(D), we have

G0α f ∂D = 0.

(c) For all α, β > 0, the resolvent equation holds: (3.4)

G0α f − G0β f + (α − β)G0α G0β f = 0,

f ∈ C(D).

(d) For any f ∈ C(D), we have (3.5)

lim αG0α f (x) = f (x),

α→+∞

x ∈ D.

Furthermore, if f |∂D = 0, then this convergence is uniform in x ∈ D, that is, (3.5′ )

lim αG0α f = f

α→+∞

in C(D).

(e) The operator G0α maps C k+θ (D) into C k+2+θ (D) for any non-negative integer k.

FELLER SEMIGROUPS AND MARKOV PROCESSES

41

(ii) (a′ ) The operator Hα , α > 0, can be uniquely extended to a non-negative, bounded linear operator on C(∂D) into C(D), denoted again by Hα , with norm kHα k ≤ 1. (b′ ) For any ϕ ∈ C(∂D), we have Hα ϕ|∂D = ϕ. (c′ ) For all α, β > 0, we have (3.6)

Hα ϕ − Hβ ϕ + (α − β)G0α Hβ ϕ = 0,

ϕ ∈ C(∂D).

(d′ ) For any ϕ ∈ C(∂D), we have lim Hα ϕ(x) = 0,

α→+∞

x ∈ D.

(e′ ) The operator Hα maps C k+2+θ (∂D) into C k+2+θ (D) for any non-negative integer k. Proof. (i) (a) Making use of Friedrichs’ mollifiers, we find that the space C θ (D) is dense in C(D) and further that non-negative functions can be approximated by non-negative smooth functions. Hence, by Lemma 3.2 it follows that the operator G0α : C θ (D) → C 2+θ (D) can be uniquely extended to a non-negative, bounded linear operator G0α : C(D) → C(D) with norm kG0α k = kG0α 1k∞ . Further, since the function G0α 1 satisfies the conditions (W − α)G0α 1 = −1 in D, G0α 1 = 0 on ∂D, by applying Theorem A.2 with W := W − α we obtain that kG0α k = kG0α 1k∞ ≤

1 . α

(b) This follows from formula (3.1), since the space C θ (D) is dense in C(D) and the operator G0α : C(D) → C(D) is bounded. (c) We find from the uniqueness theorem for problem (D) (Theorem 3.2) that equation (3.4) holds for all f ∈ C θ (D). Hence it holds for all f ∈ C(D), since the space C θ (D) is dense in C(D) and the operators G0α are bounded. (d) First, let f be an arbitrary function in C θ (D) satisfying f |∂D = 0. Then it follows from the uniqueness theorem for problem (D) (Theorem 3.2) that we have, for all α, β, f − αG0α f = G0α ((β − W )f ) − βG0α f. Thus we have, by estimate (3.3), kf − αG0α f k∞ ≤

β 1 k(β − W )f k∞ + kf k∞ , α α

so that lim kf − αG0α f k∞ = 0.

α→+∞

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KAZUAKI TAIRA

Now let f be an arbitrary function in C(D) satisfying f |∂D = 0. By means of mollifiers, we can find a sequence {fj } in C θ (D) such that fj −→ f in C(D) as j → ∞, fj = 0

on ∂D.

Then we have, by estimate (3.3), kf − αG0α f k∞ ≤ kf − fj k∞ + kfj − αG0α fj k∞ + kαG0α fj − αG0α f k∞ ≤ 2kf − fj k∞ + kfj − αG0α fj k∞ ,

and hence lim sup kf − αG0α f k∞ ≤ 2kf − fj k∞ . α→+∞

This proves assertion (3.5′ ), since kf − fj k∞ → 0 as j → ∞. To prove assertion (3.5), let f be an arbitrary function in C(D) and x an arbitrary point of D. Take a function ψ ∈ C(D) such that 0 ≤ ψ ≤ 1 on D, ψ=0 in a neighborhood of x, ψ=1 near the set ∂D. Then it follows from the non-negativity of G0α and estimate (3.3) that

(3.7)

0 ≤ αG0α ψ(x) + αG0α (1 − ψ)(x) = αG0α 1(x) ≤ 1.

However, by applying assertion (3.5′ ) to the function 1 − ψ we have lim αG0α (1 − ψ)(x) = (1 − ψ)(x) = 1.

α→+∞

In view of inequalities (3.7), this implies that lim αG0α ψ(x) = 0.

α→+∞

Thus, since −kf k∞ ψ ≤ f ψ ≤ kf k∞ ψ on D, it follows that |αG0α (f ψ)(x)| ≤ kf k∞ αG0α ψ(x) → 0

as α → +∞.

Therefore, by applying assertion (3.5′ ) to the function (1 − ψ)f we obtain that f (x) = ((1 − ψ)f ) (x) = lim αG0α ((1 − ψ)f ) (x) = lim αG0α f (x). α→+∞

α→+∞

(ii) (a′ ) Since the space C 2+θ (∂D) is dense in C (∂D), by Lemma 3.3 it follows that the operator Hα : C 2+θ (∂D) → C 2+θ (D) can be uniquely extended to a nonnegative, bounded linear operator Hα : C (∂D) → C(D). Further, by applying Theorem A.2 with W := W − α we have kHα k = kHα 1k∞ = 1.

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43

(b′ ) This follows from formula (3.2), since the space C 2+θ (∂D) is dense in C (∂D) and the operator Hα : C (∂D) → C(D) is bounded. (c′ ) We find from the uniqueness theorem for problem (D) that formula (3.6) holds for all ϕ ∈ C 2+θ (∂D). Hence it holds for all ϕ ∈ C (∂D), since the space C 2+θ (∂D) is dense in C (∂D) and the operators G0α and Hα are bounded. The proof of Theorem 3.4 is now complete. Now we consider the following boundary value problem (∗) in the framework of the spaces of continuous functions: (α − W )u = f in D, (∗) Lu = 0 on ∂D. To do this, we introduce three operators associated with problem (∗). (I) First, we introduce a linear operator W : C(D) −→ C(D) as follows. (a) The domain D(W ) of W is the space C 2+θ (D). (b) W u = P u + Sr u for all u ∈ D(W ). Then we have the following (cf. [Ta1, Lemma 9.6.5]): Lemma 3.5. The operator W has its minimal closed extension W in the space C(D). Proof. We apply part (i) of Theorem 1.5 to the operator W . Assume taht a function u ∈ C 2 (D) takes a positive maximum at a point x0 of D: u(x0 ) = max u(x) > 0. x∈D

Then it follows that ∂u (x0 ) = 0, 1 ≤ i ≤ N, ∂xi N X ∂ 2u aij (x0 ) (x0 ) ≤ 0, ∂x ∂x i j i,j=1 since the matrix (aij (x)) is positive definite. Hence we have P u(x0 ) =

N X

aij (x0 )

i,j=1

Sr u(x0 ) =

Z

D

∂ 2u (x0 ) + c(x0 )u(x0 ) ≤ 0, ∂xi ∂xj

s(x0 , y) (u(y) − u(x0 )) dy ≤ 0.

This implies that the operator W = P + Sr satisfies condition (β) of Theorem 1.5 with K0 := D and K := D. Therefore, Lemma 3.5 follows from an application of the same theorem.

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KAZUAKI TAIRA

Remark 3.6. Since the injection: C(D) → D ′ (D) is continuous, we have the formula N X

N X ∂ 2u ∂u W u(x) = a (x) (x) + bi (x) (x) + c(x)u(x) ∂x ∂x ∂x i j i i,j=1 i=1 Z N X ∂u + s(x, y) u(y) − u(x) − (x) dy, (yj − xj ) ∂x j D j=1 ij

where the right-hand side is taken in the sense of distributions. The extended operators G0α : C(D) −→ C(D) and Hα : C(∂D) −→ C(D), α > 0, still satisfy formulas (3.1) and (3.2) respectively in the following sense (cf. [Ta1, Lemma 9.6.7 and Corollary 9.6.8]): Lemma 3.7. (i) For any f ∈ C(D), we have

G0α f ∈ D(W ),

(αI − W )G0α f = f

in D.

(ii) For any ϕ ∈ C(∂D), we have

Hα ϕ ∈ D(W ),

(αI − W )Hα ϕ = 0

in D.

Here D(W ) is the domain of the closed extension W . Proof. (i) Choose a sequence {fj } in C θ (D) such that fj → f in C(D) as j → ∞. Then it follows from the boundedness of G0α that G0α fj −→ G0α f

in C(D),

and also (α − W )G0α fj = fj −→ f Hence we have

G0α f ∈ D(W ),

(αI − W )G0α f = f

in C(D).

in D.

since the operator W : C(D) → C(D) is closed. (ii) Similarly, part (ii) is proved, since the space C 2+θ (∂D) is dense in C(∂D) and the operator Hα : C(∂D) → C(D) is bounded. Corollary 3.8. Every u in D(W ) can be written in the following form: (3.8)

u = G0α (αI − W )u + Hα (u|∂D ),

α > 0.

Proof. We let w = u − G0α (αI − W )u − Hα (u|∂D ).

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45

Then it follows from Lemma 3.7 that the function w is in D(W ) and satisfies (αI − W )w = 0 in D, w=0

on ∂D.

Thus, in view of Remark 3.6 we can apply Theorem 3.1 to obtain that w = 0. This proves formula (3.8). (II) Secondly, we introduce a linear operator LG0α : C(D) −→ C(∂D) as follows. 0 (a) The domain D LG of LG0α is the space C θ (D). α (b) LG0α f = L G0α f for all f ∈ D LG0α . Then we have the following (cf. [Ta1, Lemma 9.6.9]): Lemma 3.9. The operator LG0α , α > 0, can be uniquely extended to a nonnegative, bounded linear operator LG0α : C(D) −→ C(∂D).

Proof. Let f be an arbitrary function in D(LG0α ) such that f ≥ 0 on D. Then we have 0 2 Gα f ∈ C (D), G0α f ≥ 0 on D, 0 Gα f = 0 on ∂D, and hence

∂ (G0 f ) − δ(x′ )AG0α f ∂n α ∂ = µ(x′ ) (G0α f ) + δ(x′ )f ≥ 0 on ∂D. ∂n

LG0α f = µ(x′ )

This proves that the operator LG0α is non-negative. By the non-negativity of LG0α , we have, for all f ∈ D(LG0α ), −LG0α kf k∞ ≤ LG0α f ≤ LG0α kf k∞

on ∂D.

This implies the boundedness of LG0α with norm kLG0α k = kLG0α 1k∞ . Recall that the space C θ (D) is dense in C(D) and that non-negative functions can be approximated by non-negative C ∞ functions. Hence we find that the operator LG0α can be uniquely extended to a non-negative, bounded linear operator LG0α : C(D) → C(∂D). The next lemma states a fundamental relationship between the operators LG0α and LG0β for α, β > 0 (cf. [Ta1, Lemma 9.6.10]):

46

KAZUAKI TAIRA

Lemma 3.10. For any f ∈ C(D), we have LG0α f − LG0β f + (α − β)LG0α G0β f = 0,

(3.9)

α, β > 0.

Proof. Choose a sequence {fj } in C θ (D) such that fj → f in C(D) as j → ∞. Then, by using the resolvent equation (3.4) with f := fj we have LG0α fj − LG0β fj + (α − β)LG0α G0β fj = 0. Hence formula (3.9) follows by letting j → ∞, since the operators LG0α , LG0β and G0β are all bounded. (III) Finally, we introduce a linear operator LHα : C(∂D) −→ C(∂D) as follows. (a) The domain D (LHα ) of LHα is the space C 2+θ (∂D). (b) LHα ψ = L (Hα ψ) for all ψ ∈ D (LHα ). Then we have the following (cf. [Ta1, Lemma 9.6.11]):

Lemma 3.11. The operator LHα , α > 0, has its minimal closed extension LHα in the space C(∂D). Proof. We apply part (i) of Theorem 1.5 to the operator LHα . To do this, it suffices to show that the operator LHα satisfies condition (β ′ ) with K := ∂D (or condition (β) with K := K0 = ∂D) of the same theorem. Assume that a function ψ in D(LHα ) = C 2+θ (∂D) takes its positive maximum at some point x′ of ∂D. Since the function Hα ψ is in C 2+θ (D) and satisfies

(W − α)Hα ψ = 0 Hα ψ = ψ

in D, on ∂D,

by applying Theorem A.1 (the weak maximum principle) with W := W − α to the function Hα ψ we find that the function Hα ψ takes its positive maximum at x′ ∈ ∂D. Thus we can apply Lemma A.2 with Σ3 := ∂D to obtain that ∂ (Hα ψ)(x′ ) < 0. ∂n

(3.10) Hence we have LHα ψ(x′ ) =

N−1 X

i,j=1

αij (x′ )

∂ ∂ 2ψ (x′ ) + µ(x′ ) (Hα ψ)(x′ ) ∂xi ∂xj ∂n

+ γ(x′ )ψ(x′ ) − αδ(x′ )ψ(x′ )

≤ 0.

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47

This verifies condition (β ′ ) of Theorem 1.5. Therefore, Lemma 3.11 follows from an application of the same theorem. Remark 3.12. The operator LHα enjoys the following property: (3.11) If a function ψ in the domain D LHα takes its positive maximum at some point x′ of ∂D, then we have LHα ψ(x′ ) ≤ 0. The next lemma states a fundamental relationship between the operators LHα and LHβ for α, β > 0 (cf. [Ta1, Lemma 9.6.13]): Lemma 3.13. The domain D LHα of LHα does not depend on α > 0; so we denote by D the common domain. Then we have (3.12)

LHα ψ − LHβ ψ + (α − β)LG0α Hβ ψ = 0,

α, β > 0, ψ ∈ D.

Proof. Let ψ be an arbitrary function in D(LHβ ), and choose a sequence {ψj } in D(LHβ ) = C 2+θ (∂D) such that ψj −→ ψ in C(∂D), LHβ ψj −→ LHβ ψ

in C(∂D).

Then it follows from the boundedness of Hβ and LG0α that LG0α (Hβ ψj ) = LG0α (Hβ ψj ) −→ LG0α (Hβ ψ) in C(∂D). Therefore, by using formula (3.6) with ϕ := ψj we obtain that LHα ψj = LHβ ψj − (α − β)LG0α (Hβ ψj )

−→ LHβ ψ − (α − β)LG0α (Hβ ψ) in C(∂D).

This implies that

ψ ∈ D(LHα ),

LHα ψ = LHβ ψ − (α − β)LG0α (Hβ ψ).

Conversely, by interchanging α and β we have D(LHα ) ⊂ D(LHβ ), and so D(LHα ) = D(LHβ ). This proves the lemma. Now we can give a general existence theorem for Feller semigroups on ∂D in terms of boundary value problem (∗). The next theorem tells us that the operator LHα is the infinitesimal generator of some Feller semigroup on ∂D if and only if problem (∗) is solvable for sufficiently many functions ϕ in the space C(∂D) (cf. [Ta1, Theorem 9.6.15]):

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KAZUAKI TAIRA

Theorem 3.14. (i) If the operator LHα , α > 0, is the infinitesimal generator of a Feller semigroup on ∂D, then, for each constant λ > 0, the boundary value problem

(∗′ )

(α − W )u = 0 (λ − L)u = ϕ

in D, on ∂D

has a solution u ∈ C 2+θ (D) for any ϕ in some dense subset of C(∂D). (ii) Conversely, if, for some constant λ ≥ 0, problem (∗′ ) has a solution u ∈ 2+θ C (D) for any ϕ in some dense subset of C(∂D), then the operator LHα is the infinitesimal generator of some Feller semigroup on ∂D. Proof. (i) If the operator LHα generates a Feller semigroup on ∂D, by applying part (i) of Theorem 1.5 with K := ∂D to the operator LHα we obtain that R λI − LHα = C(∂D)

for each λ > 0.

This implies that the range R (λI − LHα ) is a dense subset of C(∂D) for each λ > 0. However, if ϕ ∈ C(∂D) is in the range R (λI − LHα ), and if ϕ = (λI − LHα ) ψ with ψ ∈ C 2+θ (∂D), then the function u = Hα ψ ∈ C 2+θ (D) is a solution of problem (∗)0 . This proves part (i). (ii) We apply part (ii) of Theorem 1.5 with K := ∂D to the operator LHα . To do this, it suffices to show that the operator LHα satisfies condition (γ) of the same theorem, since it satisfies condition (β ′ ), as is shown in the proof of Lemma 3.11. By the uniqueness theorem for problem (D′ ), it follows that any function u ∈ C 2+θ (D) which satisfies the equation (α − W )u = 0

in D

can be written in the form: u = Hα (u|∂D ) ,

u|∂D ∈ C 2+θ (∂D) = D (LHα ) .

Thus we find that if there exists a solution u ∈ C 2+θ (D) of problem (∗)0 for a function ϕ ∈ C(∂D), then we have (λI − LHα ) (u|∂D ) = ϕ, and so ϕ ∈ R (λI − LHα ) . Therefore, if, for some constant λ ≥ 0, problem (∗)0 has a solution u ∈ C 2+θ (D) for any ϕ in some dense subset of C(∂D), then the range R (λI − LHα ) is dense in C(∂D). This verifies condition (γ) (with α0 := λ) of Theorem 1.5. Hence part (ii) follows from an application of the same theorem. Theorem 3.14 is proved. We conclude this subsection by giving a precise meaning to the boundary conditions Lu for functions u in the domain D(W ).

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We let D(L) = u ∈ D(W ) : u|∂D ∈ D ,

where D is the common domain of the operators LHα , α > 0. It should be noticed that the domain D(L) contains the space C 2+θ (D), since C 2+θ (∂D) = D (LHα ) ⊂ D. Corollary 3.8 tells us that every function u in D(L) ⊂ D(W ) can be written in the form (3.8) u = G0α (αI − W )u + Hα (u|∂D ) , α > 0. Then we define

Lu = LG0α (αI − W )u + LHα (u|∂D ) .

(3.13)

The next lemma justifies definition (3.13) of Lu for all u ∈ D(L) (cf. [Ta1, Lemma 9.6.16]): Lemma 3.15. The right-hand side of formula (3.13) depends only on u, not on the choice of expression (3.8). Proof. Assume that u = G0α = G0β

αI − W u + Hα (u|∂D ) βI − W u + Hβ (u|∂D ) ,

where α > 0, β > 0. Then it follows from formula (3.9) with f := αI − W u and formula (3.12) with ψ := u|∂D that LG0α αI − W u + LHα (u|∂D ) (3.14) = LG0β αI − W u − (α − β)LG0α G0β αI − W u + LHβ (u|∂D ) − (α − β)LG0α Hβ (u|∂D ) = LG0β (βI − W )u + LHβ (u|∂D ) n o + (α − β) LG0β u − LG0α G0β αI − W u − LG0α Hβ (u|∂D ) .

However, we obtain from formula (3.9) with f := u that (3.15) LG0β u − LG0α G0β αI − W u − LG0α Hβ (u|∂D ) = LG0β u − LG0α G0β βI − W u + Hβ (u|∂D ) + (α − β)G0β u = LG0β u − LG0α u − (α − β)LG0α G0β u

= 0.

Therefore, by combining formulas (3.14) and (3.15) we have LG0α αI − W u + LHα (u|∂D ) = LG0β βI − W u + LHβ (u|∂D ) . This proves Lemma 3.15.

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KAZUAKI TAIRA

3.2 End of Proof of Theorem 1. The next theorem proves Theorem 1: Theorem 3.16. We define a linear operator A : C(D) −→ C(D) as follows (see formula (0.4)). (a) The domain D(A) of A is the set D(A) = u ∈ D(W ) : u|∂D ∈ D, Lu = 0 ,

(3.16)

where D is the common domain of the operators LHα , α > 0. (b) Au = W u for all u ∈ D(A). If the boundary condition L is transversal on the boundary ∂D, then the operator A is the infinitesimal generator of some Feller semigroup on D, and the Green −1 operator Gα = (αI − A) , α > 0, is given by the formula (3.17)

−1 Gα f = G0α f − Hα LHα LG0α f ,

f ∈ C(D).

Remark 3.17. Intuitively, formula (3.17) asserts that if the boundary condition L is transversal on the boundary ∂D, then we can “piece together” a Markov process (Feller semigroup) on the boundary ∂D with W -diffusion in the interior D to construct a Markov process (Feller semigroup) on the closure D = D ∪ ∂D (see Figure 0.8). Proof of Theorem 3.16. We apply part (ii) of Theorem 1.3 to the operator A defined by formula (3.16). The proof is divided into several steps. (1) We let L0 u(x′ ) N−1 X ∂u ′ ∂ 2u ′ (x ) + β i (x′ ) (x ) = α (x ) ∂x ∂x ∂x i j i i=1 i,j=1 N−1 X

ij

′

∂u + γ(x′ )u(x′ ) + µ(x′ ) (x′ ) ∂n Z N−1 X ∂u ′ ′ (x ) dy + r(x′ , y ′ ) u(y ′ ) − u(x′ ) − (yj − xj ) ∂xj ∂D j=1 Z N−1 X ∂u ′ + t(x′ , y) u(y) − u(x′ ) − (yj − xj ) (x ) dy, ∂xj D j=1

and consider the term −δ(x′ )(W u|∂D ) in Lu as a term of “perturbation” of L0 u: Lu = L0 u − δ(x′ ) (W u|∂D ) .

FELLER SEMIGROUPS AND MARKOV PROCESSES

51

It is easy to see that the operator LHα can be decomposed as follows: LHα = L0 Hα − αδ(x′ )I. First, we prove that For all α > 0, the operator L0 Hα generates a Feller semigroup on the boundary ∂D. To do this, we remark that L0 Hα ϕ(x′ ) =

N−1 X

αij (x′ )

i,j=1

N−1 X ∂ 2ϕ ∂ϕ ′ β i (x′ ) (x′ ) + (x ) ∂xi ∂xj ∂xi i=1

∂ + γ(x′ )ϕ(x′ ) + µ(x′ ) (Hα ϕ)(x′ ) ∂n Z N−1 X ∂ϕ ′ ′ (yj − xj ) + r(x′ , y ′ ) ϕ(y ′ ) − ϕ(x′ ) − (x ) dy ∂xj ∂D j=1 Z N−1 X ∂ϕ ′ (x ) dy. + t(x′ , y) Hα ϕ(y) − ϕ(x′ ) − (yj − xj ) ∂x j D j=1 However, we have the following results: (a) The operator N−1 X ∂ϕ ′ ∂ 2ϕ ′ β i (x′ ) (x ) + (x ) + γ(x′ )ϕ(x′ ) α (x ) ϕ(x ) 7−→ ∂x ∂x ∂x i j i i=1 i,j=1 N−1 X

′

ij

′

is a second-order degenerate elliptic differential operator on ∂D with non-positive principal symbol, and γ(x′ ) ≤ 0 on ∂D. (b) The operator ∂ ϕ(x′ ) 7−→ µ(x′ ) (Hα ϕ)(x′ ) ∂n is a classical, pseudo-differential operator of order 1 on ∂D (see [Ho], [RS]). Moreover, it should be noticed that (3.18)

x′0 ∈ ∂D, ϕ ∈ C 2 (∂D), ϕ ≥ 0 on ∂D and x′0 6∈ supp ϕ ∂ =⇒ (Hα ϕ)(x′0 ) ≥ 0. ∂n

Indeed, if we let u = Hα ϕ,

52

KAZUAKI TAIRA

then we have

(α − W )u = 0 in D,

u=ϕ

on ∂D.

Since ϕ ≥ 0 on ∂D, it follows from an application of the maximum principle that Hα ϕ = u ≥ 0 in D. Hence this implies that

∂ (Hα ϕ)(x′0 ) ≥ 0, ∂n

since ϕ(x′0 ) = 0. (c) The operator ϕ(x′ ) 7−→

Z

∂D

r(x′ , y ′ ) ϕ(y ′ ) − ϕ(x′ ) −

N−1 X j=1

(yj − xj )

∂ϕ ′ ′ (x ) dy ∂xj

is a classical, pseudo-differential operator of order 2 − κ1 on ∂D. (d) The operator Z N−1 X ∂ϕ ′ (x ) dy ϕ(x′ ) 7−→ t(x′ , y) Hα ϕ(y) − ϕ(x′ ) − (yj − xj ) ∂x j D j=1

is a classical, pseudo-differential operator of order 2 −κ2 on the boundary ∂D, since 2−κ2 T ∈ L1,0 (RN ) has the transmission property with respect to the boundary ∂D (see [Bo], [RS]). (e) It should be noticed that x′0 ∈ ∂D, ϕ ∈ C 2 (∂D), ϕ ≥ 0 on ∂D and x′0 6∈ supp ϕ =⇒ L0 Hα ϕ(x′0 ) ≥ 0.

Indeed, we have, by assertion (3.18) L0 Hα ϕ(x′0 )

=

µ(x′0 ) +

Z

D

≥ 0.

∂ (Hα ϕ)(x′0 ) + ∂n

Z

∂D

r(x′0 , y ′ )ϕ(y ′ ) dy ′

t(x′0 , y)Hα ϕ(y) dy

By virtue of Theorem 2.6, this proves that the operator L0 Hα may be written in the form (2.8). (f) Finally, since the function Hα 1 takes its positive maximum 1 only on the boundary ∂D, it follows from an application of the boundary point lemma (see Appendix, Lemma A.2) that (3.19)

∂ (Hα 1) < 0 on ∂D. ∂n

FELLER SEMIGROUPS AND MARKOV PROCESSES

53

Hence we have, by transversality condition (0.3), L0 Hα 1(x′ ) ∂ = γ(x ) + µ(x ) (Hα 1)(x′ ) + ∂n ≤ 0 on ∂D. ′

′

Z

D

t(x′ , y) [Hα 1(y) − 1] dy

Thus, by applying Theorem 2.1 to the operator L0 Hα we find that (3.20)

If λ > 0 is sufficiently large, then the range R(L0 Hα − λI)

contains the space C 2+θ (∂D).

This implies that the range R(L0 Hα −λI) is a dense subset of C(∂D). Therefore, by applying part (ii) of Theorem 3.14 to the operator L0 we obtain that the operator L0 Hα is the infinitesimal generator of some Feller semigroup on ∂D, for any α > 0. (2) Next we prove that For all α > 0, the operator LHα = L0 Hα − αδ(x′ )I generates a Feller semigroup on ∂D. It should be noticed that the operator −αδ(x′ )I is a bounded linear operator on the space C(∂D) into itself, and satisfies condition (β ′ ) of Theorem 1.5, since α > 0 and δ(x′ ) ≥ 0 on ∂D. Therefore, by applying Corollary 1.6 with A = L0 H α ,

C = −αδ(x′ )I,

we obtain that the operator LHα = L0 Hα − αδ(x′ )I is the infinitesimal generator of a Feller semigroup on ∂D, for any α > 0. (3) Now we prove that (3.21)

The equation LHα ψ = ϕ has a unique solution ψ in D(LHα ) for any ϕ ∈ C(∂D); hence the inverse LHα

−1

of LHα can be defined on the whole space C(∂D).

Further the operator −LHα

−1

is non-negative and bounded on C(∂D).

We have, by inequality (3.19) and transversality condition (0.3), LHα 1(x′ ) ∂ = γ(x ) + µ(x ) (Hα 1)(x′ ) − αδ(x′ ) + ∂n < 0 on ∂D, ′

′

Z

D

t(x′ , y) [Hα 1(y) − 1] dy

54

KAZUAKI TAIRA

and so ℓα = − sup LHα 1(x′ ) > 0. x′ ∈∂D

Further, by using Corollary 1.4 with K := ∂D, A := LHα and c := ℓα we obtain that the operator LHα + ℓα I is the infinitesimal generator of some Feller semigroup on ∂D. Therefore, since ℓα > 0, it follows from an application of part (i) of Theorem 1.3 with A := LHα + ℓα I that the equation −LHα ψ = ℓα I − (LHα + ℓα I) ψ = ϕ

has a unique solution ψ ∈ D(LHα ) for any ϕ ∈ C(∂D), and further that the −1 −1 is non-negative and bounded on the operator −LHα = ℓα I − (LHα + ℓα I) space C(∂D) with norm

−1 −1

−LHα = ℓα I − (LHα + ℓα I)

∞

≤

1 . ℓα

(4) By assertion (3.21), we can define the right-hand side of formula (3.17) for all α > 0. We prove that Gα = (αI − A)

(3.22)

−1

,

α > 0.

In view of Lemmas 3.7 and 3.13, it follows that we have, for all f ∈ C(D),

and that

−1 0 0f G f = G f − H LH LG ∈ D(W ), α α α α α −1 LG0α f ∈ D LHα = D, Gα f |∂D = −LHα LG f = LG0 f − LH LH −1 LG0 f = 0, α α α α α (αI − W )Gα f = f.

This proves that

Gα f ∈ D(A),

(αI − A)Gα f = f,

that is, (αI − A)Gα = I

on C(D).

Therefore, in order to prove formula (3.22) it suffices to show the injectivity of the operator αI − A for α > 0. Assume that u ∈ D(A) and (αI − A)u = 0. Then, by Corollary 3.8, the function u can be written as u = Hα (u|∂D ) ,

u|∂D ∈ D = D LHα .

FELLER SEMIGROUPS AND MARKOV PROCESSES

55

Thus we have LHα (u|∂D ) = Lu = 0. In view of assertion (3.21), this implies that u|∂D = 0, so that u = Hα (u|∂D ) = 0

in D.

(5) The non-negativity of Gα , α > 0, follows immediately from formula (3.17), −1 since the operators G0α , Hα , −LHα and LG0α are all non-negative. (6) We prove that the operator Gα is bounded on the space C(D) with norm kGα k ≤

(3.23)

1 , α

α > 0.

To do this, it suffices to show that (3.23′ )

Gα 1 ≤

1 α

on D.

since Gα is non-negative on C(D). First, it follows from the uniqueness property of solutions of problem (D′ ) that αG0α 1 + Hα 1 = 1 + G0α (W 1) on D.

(3.24)

In fact, the both sides have the same boundary value 1 and satisfy the same equation: (α − W )u = α in D. By applying the operator L to the both hand sides of equality (3.24), we obtain that − LHα 1(x′ )

= −L1(x′ ) − LG0α (W 1)(x′ ) + αLG0α 1(x′ ) Z ′ ′ ∂ 0 ′ = −γ(x ) − µ(x ) (Gα (W 1))(x ) − t(x′ , y)G0α (W 1)(y)dy + αLG0α 1(x′ ) ∂n D ≥ αLG0α 1(x′ ) on ∂D,

since we have the assertions W 1(x) = P 1(x) + Sr 1(x) = c(x) +

Z

D

G0α (W 1) ≤ 0

on D,

G0α (W 1)|∂D = 0

s(x, y) [1 − σ(x, y)] dy ≤ 0 in D,

on ∂D.

Hence we have, by the non-negativity of −LHα (3.25)

−LHα

−1

1 LG0α 1 ≤ α

−1

, on ∂D.

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KAZUAKI TAIRA

By using formula (3.17) with f := 1, inequality (3.25) and equality (3.24), we obtain that 1 −1 0 0 LGα 1 ≤ G0α 1 + Hα 1 Gα 1 = Gα 1 + Hα −LHα α 1 0 1 = + Gα (W 1) α α 1 ≤ on D, α since the operators Hα and G0α are non-negative and since W 1 ≤ 0 in D. (7) Finally, we prove that (3.26)

The domain D(A) is everywhere dense in the space C(D).

(7-1) Before the proof, we need some lemmas on the behavior of G0α , Hα and −1 −LHα as α → +∞ (see [BCP, Proposition III.1.6]; [Ta1, Lemmas 9.6.19 and 9.6.20]): Lemma 3.18. For all f ∈ C(D), we have (3.27) lim αG0α f + Hα (f |∂D ) = f α→+∞

in C(D).

Proof. Choose a constant β > 0 and let g = f − Hβ (f |∂D ). Then, by using formula (3.6) with ϕ := f |∂D we obtain that (3.28) αG0α g − g = αG0α f + Hα (f |∂D ) − f − βG0α Hβ (f |∂D ). However, we have, by estimate (3.3),

lim G0α Hβ (f |∂D ) = 0

α→+∞

in C(D),

and by assertion (3.5′ ): lim αG0α g = g

α→+∞

in C(D),

since g|∂D = 0. Therefore, formula (3.27) follows by letting α → +∞ in formula (3.28). Lemma 3.19. The function

∂ (Hα 1) ∂n diverges to −∞ uniformly and monotonically as α → +∞. Proof. First, formula (3.6) with ϕ := 1 gives that Hα 1 = Hβ 1 − (α − β)G0α Hβ 1.

FELLER SEMIGROUPS AND MARKOV PROCESSES

57

Thus, in view of the non-negativity of G0α and Hα it follows that α ≥ β =⇒ Hα 1 ≤ Hβ 1 on D. Since Hα 1|∂D = Hβ 1|∂D = 1, this implies that the functions ∂ (Hα 1) ∂n are monotonically non-increasing in α. Further, by using formula (3.5) with f := Hβ 1 we find that the function

β Hα 1(x) = Hβ 1(x) − 1 − α

αG0α Hβ 1(x)

converges to zero monotonically as α → +∞, for each interior point x of D. Now, for any given constant K > 0 we can construct a function u ∈ C 2 (D) such that (3.29a) (3.29b)

u = 1 on ∂D, ∂u ≤ −K on ∂D. ∂n

Indeed, it follows from Theorem 3.1 that, for any integer m > 0, the function m

u = (Hα0 1) ,

α0 > 0,

belongs to C 2+θ (D) and satisfies condition (3.29a). Further we have ∂ ∂u =m (Hα0 1) ∂n ∂n ∂ ≤ m sup (Hα0 1) (x′ ). x′ ∈∂D ∂n m

In view of inequality (3.19), this implies that the function u = (Hα0 1) satisfies condition (3.29b) for m sufficiently large. Take a function u ∈ C 2 (D) which satisfies conditions (3.29a) and (3.29b), and choose a neighborhood U of ∂D, relative to D, with smooth boundary ∂U such that (see Figure 3.1) (3.30)

u≥

1 2

on U .

58

KAZUAKI TAIRA

...................................................................... ................ ............ . . . . . . . . . . .......... .... . . . . . . ........ . .... . . .. .... .... .... .... .... .... .... .... .... .... . . ....... . . . . . . . . . . . .... .... . . . . . . . . . . . .... .. ..... . .. ... . .... . ... ..... .. ... . ∂D ....... . . .. ... .. . .. . . .... . . . . .. . ... . . . ... ... .... .... U . . .. .. ... . . .... .... . ... . .. . ... ... . ... .. .. .... .... .... .. . . . . ... . .. .... ... ... .. ... . . . . . . . . . . . . . .. . . . U . . .. . . . . . . . . ... ... U . . .. .... ... ..... ... .. ... . .... . . . . . . . .. .. .... .. ... ... .... .. .. ... .. .. . ... . . . . ... . . . . . . . .... ... ... ... ... .. ... .... .... ... .. . . .... .... .... ... .... ...... ... ... . . . . ... . . . . ... .. .... .... . .... .... ... .. ∂D .... . .... .... . . . .... . . ∂D . .. .... .... .... .... .... ... .... ..... .... . . . . .... . . . . .. ... ... ...... ... .... .. . .... .. .... .... .... ....... ...... .... .... .... ... .. .... .. . . . .... .... .... .... .... .... .. . . ......... . ..... .......... ......... . . . ............. . . . . . . . . ..... ..................... ....................................................... Figure 3.1 Recall that the function Hα 1 converges to zero in D monotonically as α → +∞. Since u|∂D = Hα 1|∂D = 1, by using Dini’s theorem we can find a constant α > 0 (depending on u and hence on K) such that (3.31a) (3.31b)

Hα 1 ≤ u on ∂U \ ∂D,

α > 2kAuk∞ .

It follows from inequalities (3.30) and (3.31b) that (W − α)(Hα 1 − u) = αu − W u α ≥ − kW uk∞ 2 > 0 in U . Thus, by applying Theorem A.1 with W := W −α to the function Hα 1−u we obtain that the function Hα 1 − u may take its positive maximum only on the boundary ∂U . However, conditions (3.29a) and (3.31a) imply that Hα 1 − u ≤ 0 on ∂U = (∂U \ ∂D) ∪ ∂D. Therefore, we have Hα 1 ≤ u on U = U ∪ ∂U , and hence

∂ ∂u (Hα 1) ≤ ≤ −K ∂n ∂n

since u|∂D = Hα 1|∂D = 1. The proof of Lemma 3.19 is complete.

on ∂D,

FELLER SEMIGROUPS AND MARKOV PROCESSES

59

Corollary 3.20. If the boundary condition L is transversal on the boundary ∂D, then we have −1 lim k − LHα k = 0. α→+∞

Proof. We recall that LHα 1(x′ ) Z ∂ ′ ′ = γ(x ) + µ(x ) (Hα 1)(x ) − αδ(x ) + t(x′ , y) [Hα 1(y) − 1] dy ∂n D Z ′ ∂ ′ ′ ≤ µ(x ) (Hα 1)(x ) − αδ(x ) + t(x′ , y) [Hα 1(y) − 1] dy. ∂n D ′

′

However, it follows from an application of Beppo-Levi’s theorem that Z Z ′ lim t(x , y) [Hα 1(y) − 1] dy = − t(x′ , y) dy, α→+∞

D

D

since the function Hα 1 converges to zero in D monotonically as α → +∞. Hence we obtain from Lemma 3.19 that if the boundary condition L is transversal on the boundary ∂D, that is, if we have Z t(x′ , y)dy = +∞ if µ(x′ ) = δ(x′ ) = 0, D

then the function LHα 1 diverges to −∞ monotonically as α → +∞. By Dini’s theorem, this convergence is uniform in x′ ∈ ∂D. Hence the function 1 LHα 1(x′ ) converges to zero uniformly in x′ ∈ ∂D as α → +∞. This gives that

−1 −1

= −LH 1 −LH

α α ∞

1

≤

LHα 1 −→ 0 as α → +∞, ∞

since we have

−LHα 1(x′ ) 1

(−LHα 1(x′ )) , 1= ≤

′ |LHα 1(x )| LHα 1 ∞

x′ ∈ ∂D.

(7-2) Proof of assertion (3.26) In view of formula (3.22) and inequality (3.23), it suffices to prove that (3.32)

lim kαGα f − f k∞ = 0,

α→+∞

since the space C 2+θ (D) is dense in C(D).

f ∈ C 2+θ (D),

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KAZUAKI TAIRA

First, we remark that kαGα f − f k∞

−1

0

0 = αGα f − αHα LHα LGα f − f ∞

≤ αG0α f + Hα (f |∂D ) − f ∞

−1

+ −αHα LHα LG0α f − Hα (f |∂D ) ∞

0

≤ αGα f + Hα (f |∂D ) − f ∞

−1

+ −αLHα LG0α f − f |∂D . ∞

Thus, in view of formula (3.27) it suffices to show that i h −1 0 LGα f − f |∂D = 0 in C(∂D). (3.33) lim −αLHα α→+∞

Take a constant β such that 0 < β < α, and write f = G0β g + Hβ ϕ,

where (cf. formula (3.8)):

g = (β − W )f ∈ C θ (D),

ϕ = f |∂D ∈ C 2+θ (∂D).

Then, by using equations (3.4) (with f := g) and (3.6) we obtain that G0α f = G0α G0β g + G0α Hβ ϕ =

1 G0β g − G0α g + Hβ ϕ − Hα ϕ . α−β

Hence we have

−1

0 LGα f − f |∂D

−αLHα ∞

α

α −1 0 0 LGβ g − LGα g + LHβ ϕ + = ϕ − ϕ

α − β −LHα

α−β ∞

α −1 ≤

−LHα · LG0β g + LHβ ϕ ∞ α−β

β α −1

kϕk∞ . +

−LHα · LG0α ∞ · kgk∞ + α−β α−β

By Corollary 3.20, it follows that the first term on the last inequality converges to zero as α → +∞. For the second term, by using formula (3.4) with f := 1 and the non-negativity of G0β and LG0α we find that kLG0α k = kLG0α 1k∞

= kLG0β 1 − (α − β)LG0α G0β 1k∞

≤ kLG0β 1k∞ .

FELLER SEMIGROUPS AND MARKOV PROCESSES

61

Hence the second term also converges to zero as α → +∞. It is clear that the third term converges to zero as α → +∞. This completes the proof of assertion (3.33) and hence that of assertion (3.32). (8) Summing up, we have proved that the operator A, defined by formula (3.16), satisfies conditions (a) through (d) in Theorem 1.3. Hence it follows from an application of the same theorem that the operator A is the infinitesimal generator of some Feller semigroup on D. The proof of Theorem 3.16 and hence that of Theorem 1 is now complete. 4. Proof of Theorem 2 In this section we prove Theorem 2. First, we show that if condition (A) is satisfied, then the operator LHα is bijective in the framework of H¨older spaces. This is proved by applying Theorem 2.1 just as in the proof of Theorem 1. Therefore, we find that a unique solution u of problem (α − W )u = f in D, (∗∗) Lu = m(x′ ) Lν u + γ(x′ )u = 0 on ∂D can be expressed as follows: −1 ν ν u = Gα f = Gα f − Hα LHα (LGα f ) . This formula allows us to verify all the conditions of the generation theorems of Feller semigroups discussed in Subsection 1.2.

4.1 The Space C0 (D \ M ). First, we consider a one-point compactification K∂ = K ∪ {∂} of the space K = D \ M , where M = {x′ ∈ ∂D : m(x′ ) = 0}.

We say that two points x and y of D are equivalent modulo M if x = y or x, y ∈ M ; we then write x ∼ y. We denote by D/M the totality of equivalence classes modulo M . On the set D/M , we define the quotient topology induced by the projection q : D → D/M . It is easy to see that the topological space D/M is a one-point compactification of the space D \ M and that the point at infinity ∂ corresponds to the set M : K∂ = D/M, ............ ∂D .............. ........................ ...... ....... . . . . . .... ...... .... . . . . . . . ... ........... . . . . . . . ... . . . . . . . . . . . . . .. . . . . . .... ... .... ..... ... . D .. ... ... ... . . ... .. .... .... ... ..... . . . . ........ ..................................................................................

..............................

................................................

∂ = M.

........................................... ................. ...... ............ . .... . . . . . . . . ... ........ . . . ... . . . . . . ... . . . . . . . .. .... . . . . . . . . . .. D/M ... .. .... ... . ... .. .... ... ...... ... . ......... . .... ............ ...................................................................

M Figure 4.1

• ∂

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KAZUAKI TAIRA

Furthermore, we have the following isomorphism: C(K∂ ) ∼ = u ∈ C(D) : u is constant on M .

(4.1)

Now we introduce a closed subspace of C(K∂ ) as in Subsection 1.1: C0 (K) = {u ∈ C(K∂ ) : u(∂) = 0} . Then we have, by assertion (4.1), C0 (K) ∼ = C0 (D \ M ) = u ∈ C(D) : u = 0 on M .

(4.2)

4.2 End of Proof of Theorem 2. We shall apply part (ii) of Theorem 1.3 to the operator W defined by formula (0.5). First, we simplify the boundary condition Lu = 0 on ∂D. If conditions (A) and (H) are satisfied, then we may assume that the boundary condition L is of the form Lu = m(x′ ) Lν u + (m(x′ ) − 1)u,

(4.3) with

0 ≤ m(x′ ) ≤ 1

on ∂D.

Indeed, it suffices to note that the boundary condition Lu = m(x′ ) Lν u + γ(x′ ) (u|∂D ) = 0 on ∂D is equivalent to the condition

m(x′ ) m(x′ ) − γ(x′ )

Lν u +

γ(x′ ) m(x′ ) − γ(x′ )

(u|∂D ) = 0

on ∂D.

Furthermore, we remark that LG0α f = m(x′ ) Lν G0α f, and LHα ϕ = m(x′ ) Lν Hα ϕ + (m(x′ ) − 1)ϕ. Hence, in view of definition (3.13) it follows that (4.3′ )

Lu = m(x′ ) Lν u + (m(x′ ) − 1) (u|∂D ) ,

Therefore, the next theorem proves Theorem 2:

u ∈ D(L).

FELLER SEMIGROUPS AND MARKOV PROCESSES

63

Theorem 4.1. We define a linear operator W : C0 (D \ M ) −→ C0 (D \ M ) as follows (cf. formula (3.16)). (a) The domain D(W) of W is the set (4.4)

D(W) = {u ∈ C0 (D \ M ) : W u ∈ C0 (D \ M ),

Lu = m(x′ ) Lν u + (m(x′ ) − 1) (u|∂D ) = 0}.

(b) Wu = W u for all u ∈ D(W). Assume that the following condition (A′ ) is satisfied: (A′ ) 0 ≤ m(x′ ) ≤ 1 on ∂D. Then the operator W is the infinitesimal generator of some Feller semigroup {Tt }t≥0 on D \ M , and the Green operator Gα = (αI − W)−1 , α > 0, is given by the formula −1 (4.5) Gα f = Gνα f − Hα LHα (LGνα f ) , f ∈ C0 (D \ M ). Here Gνα is the Green operator for the boundary condition Lν given by formula (3.17): −1 ν 0 0 Lν Gα f , f ∈ C(D). Gα f = Gα f − Hα Lν Hα

Proof. We apply part (ii) of Theorem 1.3 to the operator W defined by formula (4.4), just as in the proof of Theorem 1. The proof is divided into several steps. (1) First, we prove that For all α > 0, the operator LHα generates a Feller semigroup on the boundary ∂D. 2−κ2 By virtue of the transmission property of T ∈ L1,0 (RN ), it follows (see [Bo], [RS, Chapter 3]) that the operator LHα is the sum of a degenerate elliptic differential operator of second order and a classical pseudo-differential operator of order 2 − min(κ1 , κ2 ):

LHα ϕ(x′ ) = m(x′ ) Lν Hα ϕ(x′ ) + (m(x′ ) − 1)ϕ(x′ ) N−1 N−1 2 X X ∂ ϕ ∂ϕ ′ i = m(x′ ) (x′ ) + (x ) αij (x′ ) β (x′ ) ∂xi ∂xj ∂xi i,j=1 i=1 + (m(x′ ) − 1) − αm(x′ ) δ(x′ ) ϕ(x′ ) ∂ (Hα ϕ) (x′ ) + m(x′ ) µ(x′ ) ∂n Z N−1 X ∂ϕ ′ ′ (yj − xj ) r(x′ , y ′ ) ϕ(y ′ ) − ϕ(x′ ) − + (x ) dy ∂x j ∂D j=1

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KAZUAKI TAIRA

+

Z

D

t(x′ , y) Hα ϕ(y) − ϕ(x′ ) −

N−1 X j=1

(yj − xj )

∂ϕ ′ (x ) dy. ∂xj

Furthermore, it follows from an application of the boundary point lemma (see Appendix, Lemma A.2) that ∂ (Hα 1) < 0 on ∂D. ∂n This implies that LHα 1(x′ ) = m(x′ ) Lν Hα 1(x′ ) + (m(x′ ) − 1) ≤ 0

on ∂D.

Thus, by applying Theorem 2.1 to the operator LHα (cf. the proof of assertion (3.20)) we obtain that (4.6)

If λ > 0 is sufficiently large, then the range R(LHα − λI)

contains the space C 2+θ (∂D).

This implies that the range R(LHα −λI) is a dense subset of C(∂D). Therefore, by applying part (ii) of Theorem 3.14 to the operator L we obtain that the operator LHα is the infinitesimal generator of some Feller semigroup on ∂D, for all α > 0. (2) Now we prove that (4.7)

If condition (A′ ) is satisfied, then the equation LHα ψ = ϕ has a unique solution ψ in D LHα for any ϕ ∈ C(∂D); hence the inverse LHα

−1

of LHα can be defined on the whole space C(∂D).

Further the operator −LHα

−1

is non-negative and bounded on C(∂D).

Since we have, by inequality (3.19) and condition (A′ ), LHα 1(x′ ) = m(x′ )Lν Hα 1(x′ ) + (m(x′ ) − 1) ∂ = µ(x′ ) (Hα 1)(x′ ) + (m(x′ ) − 1) ∂nZ + m(x′ )

D

0. x′ ∈∂D

Here it should be noticed that the constants kα are increasing in α > 0: α ≥ β > 0 =⇒ kα ≥ kβ .

FELLER SEMIGROUPS AND MARKOV PROCESSES

65

Moreover, by using Corollary 1.4 with K := ∂D, A := LHα and c := kα we obtain that the operator LHα + kα I is the infinitesimal generator of some Feller semigroup on ∂D. Therefore, since kα > 0, it follows from an application of part (i) of Theorem 1.3 with A = LHα + kα I that the equation −LHα ψ = kα I − (LHα + kα I) ψ = ϕ

has a unique solution ψ ∈ D LHα for any ϕ ∈ C(∂D), and further that the −1 −1 is non-negative and bounded on the operator −LHα = kα I − (LHα + kα I) space C(∂D) with norm

−1 1 −1

.

−LHα = kα I − (LHα + kα I)

≤ kα

(4.8)

(3) By assertion (4.7), we can define the operator Gα by formula (4.5) for all α > 0. We prove that Gα = (αI − W)−1 ,

(4.9)

α > 0.

By virtue of Lemma 3.7 and Theorem 3.16, it follows that we have, for all f ∈ C0 (D \ M ), Gα f ∈ D(W ), and W Gα f = αGα f − f. Furthermore, we have (4.10)

LGα f =

LGνα f

− LHα LHα

−1

(LGνα f )

= 0 on ∂D.

However, we recall that (4.3′ )

Lu = m(x′ ) Lν u + (m(x′ ) − 1) (u|∂D ) ,

u ∈ D(L).

Hence we find that formula (4.10) is equivalent to the following: (4.10′ )

m(x′ ) Lν (Gα f ) + (m(x′ ) − 1) (Gα f |∂D ) = 0

on ∂D.

This implies that Gα f = 0

on M = {x′ ∈ ∂D : m(x′ ) = 0},

and so W Gα f = αGα f − f = 0 on M . Summing up, we have proved that Gα f ∈ D(W) = u ∈ C0 (D \ M ) : W u ∈ C0 (D \ M ), Lu = 0 ,

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KAZUAKI TAIRA

and (αI − W)Gα f = f,

f ∈ C0 (D \ M ),

that is, (αI − W)Gα = I

on C0 (D \ M ).

Therefore, in order to prove formula (4.9) it suffices to show the injectivity of the operator αI − W for α > 0. Assume that u ∈ D(W) and (αI − W)u = 0. Then, by Corollary 3.8 it follows that the function u can be written in the form u|∂D ∈ D = D LHα .

u = Hα (u|∂D ) , Thus we have

LHα (u|∂D ) = Lu = 0. In view of assertion (4.7), this implies that u|∂D = 0, so that u = Hα (u|∂D ) = 0

in D.

(4) Now we prove the following three assertions: (i) The operator Gα is non-negative on the space C0 (D \ M ): f ∈ C0 (D \ M ), f ≥ 0

on D \ M =⇒ Gα f ≥ 0 on D \ M .

(ii) The operator Gα is bounded on the space C0 (D \ M ) with norm kGα k ≤

1 , α

α > 0.

(iii) The domain D(W) is everywhere dense in the space C0 (D \ M ). (i) First, we show the non-negativity of Gα on the space C(D): f ∈ C(D), f ≥ 0

on D =⇒ Gα f ≥ 0

on D.

Recall that the Dirichlet problem ′

(D )

(α − W )u = f u=ϕ

in D, on ∂D

is uniquely solvable. Hence it follows that (4.11)

Gνα f = Hα (Gνα f |∂D ) + G0α f

on D.

FELLER SEMIGROUPS AND MARKOV PROCESSES

67

Indeed, the both sides have the same boundary values Gνα f |∂D and satisfy the same equation: (α − W )u = f in D. Thus, by applying the operator L to the both sides of formula (4.11) we obtain that LGνα f = LHα (Gνα f |∂D ) + LG0α f. Since the operators −LHα

−1

and LG0α are non-negative, it follows that

−1 −1 ν ν 0 −LHα (LGα f ) = −Gα f |∂D + −LHα LGα f ≥ −Gνα f |∂D

on ∂D.

Therefore, by the non-negativity of Hα and G0α we find that −1 Gα f = Gνα f + Hα −LHα (LGνα f ) ≥ Gνα f − Hα (Gνα f |∂D ) = G0α f

≥ 0 on D. (ii) Next we prove the boundedness of Gα on the space C0 (D \ M ) with norm (4.12)

kGα k ≤

1 , α

α > 0.

To do this, it suffices to show that (4.12′ )

f ∈ C0 (D \ M ), f ≥ 0

on D =⇒ αGα f ≤ max f

on D,

D

since Gα is non-negative on the space C(D). We remark (cf. formula (4.3′ )) that LGνα f = m(x′ ) Lν Gνα f + (m(x′ ) − 1) (Gνα f |∂D ) = (m(x′ ) − 1) (Gνα f |∂D ) , so that (4.5′ )

−1 Gα f = Gνα f − Hα LHα (LGνα f ) −1 ν ′ ν = Gα f + Hα −LHα ((m(x ) − 1)Gα f |∂D ) . −1

Therefore, by the non-negativity of Hα and −LHα it follows that −1 Gα f = Gνα f + Hα −LHα ((m(x′ ) − 1)Gνα f |∂D ) ≤ Gνα f 1 ≤ max f α D

on D,

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KAZUAKI TAIRA

since (m − 1)Gνα f |∂D ≤ 0 on ∂D and kGνα k ≤ 1/α. This proves assertion (4.12′ ) and hence assertion (4.12). (iii) Finally, we prove the density of D(W) in the space C0 (D \ M ). In view of formula (4.9), it suffices to show that (4.13)

lim kαGα f − f k∞ = 0,

α→+∞

f ∈ C0 (D \ M ) ∩ C ∞ (D).

We recall (cf. formula (4.5′ )) that (4.14)

−1 αGα f − f = αGνα f − f − αHα LHα (LGνα f ) −1 ν ′ ν = (αGα f − f ) + Hα LHα (α(1 − m(x ))Gα f |∂D ) .

We estimate each term on the right of formula (4.14). (iii-1) First, by applying Theorem 1 to the boundary condition Lν we find from assertion (3.32) that the first term on the right of formula (4.14) tends to zero: lim kαGνα f − f k∞ = 0.

(4.15)

α→+∞

(iii-2) To estimate the second term on the right of formula (4.14), we remark that −1 Hα LHα (α(1 − m(x′ ))Gνα f |∂D ) −1 −1 ′ ′ ν = Hα LHα ((1 − m(x ))f |∂D ) + Hα LHα ((1 − m(x ))(αGα f − f )|∂D ) .

However, we have, by assertion (4.8),

−1

(4.16)

Hα LHα ((1 − m(x′ )) (αGνα f − f ) |∂D ) ∞

−1

′ ν ≤ −LHα · k(1 − m(x )) (αGα f − f ) |∂D k∞ 1 k(1 − m(x′ )) (αGνα f − f ) |∂D k∞ kα 1 kαGνα f − f k∞ −→ 0 as α → +∞. ≤ k1 ≤

Here we have used the fact: k1 = − sup LH1 1(x′ ) ≤ kα = − sup LHα 1(x′ ) for all α ≥ 1. x′ ∈∂D

x′ ∈∂D

Thus we are reduced to the study of the term −1 Hα LHα ((1 − m(x′ ))f |∂D ) .

Now, for any given ε > 0, we can find a function h ∈ C ∞ (∂D) such that h = 0 near M = {x′ ∈ ∂D : m(x′ ) = 0}, k(1 − m(x′ ))f |∂D − hk∞ < ε.

FELLER SEMIGROUPS AND MARKOV PROCESSES

69

Then we have, for all α ≥ 1,

−1 −1

(4.17)

Hα LHα ((1 − m(x′ ))f |∂D ) − Hα LHα h ∞

−1

≤ −LHα · k(1 − m(x′ ))f |∂D − hk∞ ε ≤ kα ε ≤ . k1 Furthermore, we can find a function θ ∈ C0∞ (∂D) such that θ=1 near M , (1 − θ)h = h on ∂D. Then we have h(x′ ) = (1 − θ(x′ )) h(x′ ) 1 − θ(x′ ) ′ h(x′ ) = (−LHα 1(x )) −LHα 1(x′ ) 1 − θ(x′ ) ≤ sup khk∞ (−LHα 1(x′ )) . ′ x′ ∈∂D −LHα 1(x ) Since the operator −LHα

−1

is non-negative on the space C(∂D), it follows that 1 − θ(x′ ) −1 −LHα h ≤ sup · khk∞ on ∂D, ′ x′ ∈∂D −LHα 1(x )

so that (4.18)

−1

Hα LHα h

−1

≤ −LHα h

∞

∞

≤ sup

x′ ∈∂D

1 − θ(x′ ) −LHα 1(x′ )

However, there exists a constant c0 > 0 such that 0≤

1 − θ(x′ ) ≤ c0 , m(x′ )

x′ ∈ ∂D.

Hence we have 1 − θ(x′ ) m(x′ ) (−Lν Hα 1(x′ )) + (1 − m(x′ ))

1

≤ c0

−Lν Hα 1 .

1 − θ(x′ ) ≤ −LHα 1(x′ )

∞

In view of Lemma 3.19, this implies that 1 − θ(x′ ) = 0. lim sup α→+∞ x′ ∈∂D −LHα 1(x′ )

· khk∞ .

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KAZUAKI TAIRA

Summing up, we obtain from inequalities (4.17) and (4.18) that

−1

lim sup Hα LHα ((1 − m(x′ ))f |∂D ) ∞ α→+∞ −1

≤ lim sup Hα LHα h ∞

α→+∞

−1 −1

′ + Hα LHα ((1 − m(x ))f |∂D ) − Hα LHα h ∞ ′ ε 1 − θ(x ) khk∞ + ≤ lim sup ′ α→+∞ x′ ∈∂D −LHα 1(x ) k1 ε ≤ . k1

Since ε is arbitrary, this proves that (4.19)

−1

lim Hα LHα ((1 − m(x′ ))f |∂D )

α→+∞

= 0.

∞

Therefore, by combining assertions (4.16) and (4.19) we find that the second term on the right of formula (4.14) also tends to zero:

−1

lim Hα LHα (α(1 − m(x′ ))Gνα f |∂D )

α→+∞

∞

= 0.

This completes the proof of assertion (4.13) and hence that of assertion (iii). (5) Summing up, we have proved that the operator W, defined by formula (4.4), satisfies conditions (a) through (d) in Theorem 1.3. Hence, in view of assertion (4.2) it follows from an application of part (ii) of the same theorem that the operator W is the infinitesimal generator of some Feller semigroup {Tt }t≥0 on D \ M . The proof of Theorem 4.1 and hence that of Theorem 2 is now complete. Appendix: The Maximum Principle In this appendix, following Bony–Courr`ege–Priouret [BCP] we formulate two useful maximum principles for second-order elliptic Waldenfels operators. First, we state the weak maximum principle. Theorem A.1 (The weak maximum principle). Let W be a second-order elliptic Waldenfels operator. Assume that a function u ∈ C(D) ∩ C 2 (D) satisfies either W u ≥ 0 and W 1 < 0 in D or W u > 0 and W 1 ≤ 0 in D. Then the function u may take its positive maximum only on the boundary ∂D. Next we consider the interior normal derivative (∂u)/(∂n) at a boundary point where the function u ∈ C 2 (D) takes its non-negative maximum. The boundary point lemma reads as follows:

FELLER SEMIGROUPS AND MARKOV PROCESSES

71

Lemma A.2 (The boundary point lemma). Let W be a second-order elliptic Waldenfels operator. Assume that a function u ∈ C(D) ∩ C 2 (D) satisfies W u ≥ 0 in D, and that there exists a point x′0 of ∂D such that

u(x′0 ) = maxx∈D u(x) ≥ 0,

u(x) < u(x′0 ),

x ∈ D.

Then the interior normal derivative (∂u)/(∂n)(x′0) of u at x′0 , if it exists, satisfies the condition ∂u ′ (x ) < 0. ∂n 0 For a proof of Theorem A.1 and Lemma A.2, the reader might refer to Bony– Courr`ege–Priouret [BCP] and also Taira [Ta1]. References [BCP] J.-M. Bony, P. Courr` ege et P. Priouret, Semi-groupes de Feller sur une vari´ et´ e ` a bord compacte et probl` emes aux limites int´ egro-diff´ erentiels du second ordre donnant lieu au principe du maximum, Ann. Inst. Fourier (Grenoble) 18 (1968), 369–521. [Bo] L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), 11–51. [Ca] C. Cancelier, Probl` emes aux limites pseudo-diff´ erentiels donnant lieu au principe du maximum, Comm. P.D.E. 11 (1986), 1677–1726. [CP] Chazarain, J. et A. Piriou, Introduction ` a la th´ eorie des ´ equations aux d´ eriv´ ees partielles lin´ eaires, Gauthier-Villars, Paris, 1981. [CM] R. R. Coifman et Y. Meyer, Au-del` a des op´ erateurs pseudo-diff´ erentiels, Ast´ erisque No. 57, Soc. Math. France, Paris, 1978. [Ho] L. H¨ ormander, The analysis of linear partial differential operators III, Springer-Verlag, Berlin Heidelberg New York Tokyo, 1985. [Ku] H. Kumano-go, Pseudo-differential operators, MIT Press, Cambridge, Mass., 1981. [OR] O.A. Ole˘ınik and E.V. Radkeviˇ c, Second order equations with nonnegative characteristic form, (in Russian), Itogi Nauki, Moscow, 1971; English translation, Amer. Math. Soc., Providence, Rhode Island and Plenum Press, New York, 1973. [RS] S. Rempel and B.-W. Schulze, Index theory of elliptic boundary problems, Akademie-Verlag, Berlin, 1982. [SU] K. Sato and T. Ueno, Multi-dimensional diffusion and the Markov process on the boundary, J. Math. Kyoto Univ. 14 (1965), 529–605. [Ta1] K. Taira, Diffusion processes and partial differential equations, Academic Press, San Diego New York London Tokyo, 1988. [Ta2] K. Taira, Boundary value problems and Markov processes, Lecture Notes in Math. No. 1499, Springer-Verlag, Berlin Heidelberg New York Tokyo, 1991. [Ta3] K. Taira, On the existence of Feller semigroups with boundary conditions, Memoirs Amer. Math. Soc. No. 475 (1992). [Ta4] K. Taira, On the existence of Feller semigroups with boundary conditions III, Hiroshima Math. J. 27 (1997), 77–103. [TW] S. Takanobu and S. Watanabe, On the existence and uniqueness of diffusion processes with Wentzell’s boundary conditions, J. Math. Kyoto Univ. 28 (1988), 71–80. [Ty] M. Taylor, Pseudodifferential operators, Princeton Univ. Press, Princeton, 1981. [Tr] H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. [Wa] S. Watanabe, Construction of diffusion processes with Wentzell’s boundary conditions by means of Poisson point processes of Brownian excursions, Probability Theory, Banach Center Publications, Vol. 5, PWN-Polish Scientific Publishers, Warsaw, 1979, pp. 255–271.

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[We]

[Yo]

A. D. Wentzell (Ventcel’), On boundary conditions for multidimensional diffusion processes (in Russian), Teoriya Veroyat. i ee Primen. 4 (1959), 172–185; English translation in Theory Prob. and its Appl. 4 (1959), 164–177. K. Yosida, Functional analysis, sixth edition, Springer-Verlag, Berlin Heidelberg New York, 1980.

Institute of Mathemati s, University of Tsukuba, Tsukuba 305{8571, Japan

E-mail address: [email protected]

Kazuaki TAIRA Department of Mathematics, Hiroshima University, Higashi-Hiroshima 739, Japan

This paper provides a careful and accessible exposition of the functional analytic approach to the problem of construction of Markov processes with boundary conditions in probability theory. Our approach is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. In this paper we construct a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it “dies” at the time when it reaches the set where the particle is definitely absorbed.

Abstra t.

Table of contents 0. Introduction and Results 1. Theory of Feller Semigroups 1.1 Markov Processes 1.2 Markov Transition Functions and Feller Semigroups 1.3 Generation Theorems of Feller Semigroups 2. Theory of Pseudo-Differential Operators 2.1 Function Spaces 2.2 Pseudo-Differential Operators 2.3 Unique Solvability Theorem for Pseudo-Differential Operators 3. Proof of Theorem 1 3.1 General Existence Theorem for Feller Semigroups 3.2 End of Proof of Theorem 1 4. Proof of Theorem 2 4.1 The Space C0 (D \ M ) 4.2 End of Proof of Theorem 2 Appendix: The maximum principle References 1991 Mathematics Subject Classification. Primary 47D07, 35J25; Secondary 47D05, 60J35, 60J60. Key words and phrases. Feller semigroup, Markov process, elliptic boundary value problem. Typeset by AMS-TEX 1

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KAZUAKI TAIRA

0. Introduction and Results Let D be a bounded, convex domain of Euclidean space RN with smooth boundary ∂D; its closure D = D ∪ ∂D is an N -dimensional compact smooth manifold with boundary. ∂D ...........................

................................................. .......... . ... ......•.... . . . . ..... ... . . . ...... . ... .... ..... . . .. .. . ..... ...... . ... n D .. ... . ..... ... . . ....... . . .......... ....... . . . . . . . ...................... . . . ............................................ Figure 0.1 Let C(D) be the space of real-valued, continuous functions on D. We equip the space C(D) with the topology of uniform convergence on the whole D; hence it is a Banach space with the maximum norm kf k∞ = max |f (x)|. x∈D

A strongly continuous semigroup {Tt }t≥0 on the space C(D) is called a Feller semigroup on D if it is non-negative and contractive on C(D): f ∈ C(D), 0 ≤ f ≤ 1

on D =⇒ 0 ≤ Tt f ≤ 1 on D.

It is known (see [Ta1]) that if Tt is a Feller semigroup on D, then there exists a unique Markov transition function pt on D such that Z pt (x, dy)f (y), f ∈ C(D). Tt f (x) = D

It can be shown that the function pt is the transition function of some strong Markov process; hence the value pt (x, E) expresses the transition probability that a Markovian particle starting at position x will be found in the set E at time t. Furthermore, it is known (see [BCP], [SU], [Ta1], [We]) that the infinitesimal generator of a Feller semigroup {Tt }t≥0 is described analytically by a Waldenfels operator W and a Ventcel’ boundary condition L, which we formulate precisely. Let W be a second-order elliptic integro-differential operator with real coefficients such that (0.1)

W u(x) = P u(x) + Sr u(x) N N 2 X X ∂u ∂ u (x) + bi (x) (x) + c(x)u(x) := aij (x) ∂x ∂x ∂x i j i i=1 i,j=1 Z N X ∂u + s(x, y) u(y) − σ(x, y) u(x) − (x) dy. (yj − xj ) ∂x j D j=1

FELLER SEMIGROUPS AND MARKOV PROCESSES

3

Here: (1) aij ∈ C ∞ (RN ), aij (x) = aji (x) for x ∈ boldRN and 1 ≤ i, j ≤ N , and there exists a constant a0 > 0 such that N X

i,j=1

aij (x)ξi ξj ≥ a0 |ξ|2 ,

x ∈ RN , ξ ∈ RN .

(2) bi ∈ C ∞ (RN ). (3) c ∈ C ∞ (RN ) and c(x) ≤ 0 in D. (4) The integral kernel s(x, y) is the distribution kernel of a properly supported, N pseudo-differential operator S ∈ L2−κ 1,0 (R ), κ > 0, which has the transmission property with respect to ∂D (see Subsection 2.2), and s(x, y) ≥ 0 off the diagonal {(x, x) : x ∈ RN } in RN × RN . The measure dy is the Lebesgue measure on RN . (5) The function σ(x, y) is a local unity function on D, that is, σ(x, y) is a smooth function on D × D such that σ(x, y) = 1 in a neighborhood of the diagonal ∆D = {(x, x) : x ∈ D} in D × D. The function σ(x, y) depends on the shape of the domain D. More precisely, it depends on a family of local charts on D in each of which the Taylor expansion is valid for functions u. For example, if D is convex , we may take σ(x, y) ≡ 1 on D × D. R (6) W 1(x) = P 1(x) + Sr 1(x) = c(x) + D s(x, y) [1 − σ(x, y)] dy ≤ 0 in D. The intuitive meaning of condition (6) is that the jump phenomenon from a point x ∈ D to the outside of a neighborhood of x in the interior D is “dominated” by the absorption phenomenon at x. In particular, if c(x) ≡ 0 in D, then condition (6) implies that any Markovian particle does not move by jumps from x ∈ D to the outside of a neighborhood V (x) of x in the interior D, since we have the assertion Z

D

s(x, y) [1 − σ(x, y)] dy = 0,

and so, by conditions (4) and (5), s(x, y) = 0 for all y ∈ D \ V (x).

.......................................... ................. ......... . . . . . . . D ..... ...... . ..... . . ....... . .... .... .... . . .. . ... ... . .. . ... ... .. . . . . .. . ..... . .. ... . . ... . . . . . ..... .... ... . . . . . ... ... .. . . . ... ....... .. ....... ... ........ ... . . . . . . . . . ... ..... ... ....... ... ... ... . ... ... ...... .... .... .. .. ... .... . . . ...... . ..... ....... . . . . . . .......... ............................................................. Figure 0.2

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KAZUAKI TAIRA

The operator W is called a second-order Waldenfels operator (cf. [BCP]). The differential operator P is called a diffusion operator which describes analytically a strong Markov process with continuous paths (diffusion process) in the interior D. The operator Sr is called a second-order L´evy operator which is supposed to correspond to the jump phenomenon in the interior D; a Markovian particle moves by jumps to a random point, chosen with kernel s(x, y), in the interior D. Therefore, the Waldenfels operator W is supposed to correspond to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space D (see Figure 0.2). It should be emphasized that the integral operator Sr is a “regularization” of S, since the integrand is absolutely convergent. Indeed, we can write Sr u(x) in the form Sr u(x) Z N X ∂u (x) dy = s(x, y) u(y) − σ(x, y) u(x) + (yj − xj ) ∂x j D j=1 Z = s(x, y) [1 − σ(x, y)] u(y) dy D Z N X ∂u (x) dy. + s(x, y)σ(x, y) u(y) − u(x) − (yj − xj ) ∂x j D j=1

By using Taylor’s formula u(y) − u(x) − =

N X

i,j=1

N X j=1

(yj − xj )

(yi − xi )(yj − xj )

Z

∂u (x) ∂xj

0

1

∂ 2u (x + t(y − x))dt , (1 − t) ∂xi ∂xj

we can find a constant C1 > 0 such that N X ∂u ≤ C1 |x − y|2 , u(y) − u(x) − (x) (y − x ) j j ∂x j j=1

x, y ∈ D.

On the other hand, it follows from an application of [CM, Chapitre IV, Proposition 1]) that, for any compact K ⊂ RN , there exists a constant C2 > 0 such that the N distribution kernel s(x, y) of S ∈ L2−κ 1,0 (R ), κ > 0, satisfies the estimate 0 ≤ s(x, y) ≤

C2 , |x − y|N+2−κ

x, y ∈ D, x 6= y.

Therefore, we have, with some constant C3 > 0, Z N X ∂u s(x, y)σ(x, y) u(y) − u(x) − (yj − xj ) (x) dy ∂xj D j=1

FELLER SEMIGROUPS AND MARKOV PROCESSES

≤ C3 kukC 2 (D) = C3 kukC 2 (D)

Z

ZD D

5

1 · |x − y|2 dy |x − y|N+2−κ 1 dy. |x − y|N−κ

Similarly, we have, with some constant C4 > 0, Z Z s(x, y) [1 − σ(x, y)] u(y) dy ≤ C4 kuk C(D) D

D

1 dy, |x − y|N−κ

since we have the formula σ(x, y) − 1

= σ(x, y) − σ(x, x) − =

N X

i,j=1

N X j=1

(yi − xi )(yj − xj )

(yj − xj )

Z

1 0

∂σ (x, x) ∂xj

∂ 2σ (1 − t) (x, x + t(y − x))dt . ∂xi ∂xj

Let L be a second-order boundary condition such that in local coordinates (x1 , x2 , . . . , xN−1 ) (0.2) ∂u Lu(x′ ) = Qu(x′ ) + µ(x′ ) (x′ ) − δ(x′ )W u(x′ ) + Γr u(x′ ) ∂n N−1 N−1 2 X X ∂ u ∂u ′ αij (x′ ) := β i (x′ ) (x′ ) + (x ) + γ(x′ )u(x′ ) ∂x ∂x ∂x i j i i,j=1 i=1

∂u + µ(x′ ) (x′ ) − δ(x′ )W u(x′ ) ∂n Z N−1 X ∂u ′ ′ (x ) dy + r(x′ , y ′ ) u(y ′ ) − τ (x′ , y ′ ) u(x′ ) − (yj − xj ) ∂x j ∂D j=1 Z + t(x′ , y) [u(y) − u(x′ )] dy . D

Here: (1) The operator Q is a second-order degenerate elliptic differential operator on ij ∂D with non-positive principal symbol. In other words the α are the compo2 nents of a smooth symmetric contravariant tensor of type 0 on ∂D satisfying the condition N−1 X

i,j=1

ij

′

α (x )ξi ξj ≥ 0,

′

′

x ∈ ∂D, ξ =

N−1 X

Here Tx∗′ (∂D) is the cotangent space of ∂D at x′ .

j=1

ξj dxj ∈ Tx∗′ (∂D).

6

KAZUAKI TAIRA

PN−1 (2) β(x′ ) = i=1 β i (x′ )∂u/∂xi is a smooth vector field on ∂D. (3) Q1 = γ ∈ C ∞ (∂D) and γ(x′ ) ≤ 0 on ∂D. (4) µ ∈ C ∞ (∂D) and µ(x′ ) ≥ 0 on ∂D. (5) δ ∈ C ∞ (∂D) and δ(x′ ) ≥ 0 on ∂D. (6) n = (n1 , n2 , . . . , nN ) is the unit interior normal to the boundary ∂D. (7) The integral kernel r(x′ , y ′ ) is the distribution kernel of a pseudo-differential 2−κ1 operator R ∈ L1,0 (∂D), κ1 > 0, and r(x′ , y ′ ) ≥ 0 off the diagonal ∆∂D = ′ ′ ′ {(x , x ) : x ∈ ∂D} in ∂D × ∂D. The density dy ′ is a strictly positive density on ∂D. (8) The integral kernel t(x, y) is the distribution kernel of a properly supported, 1−κ2 (RN ), κ2 > 0, which has the transmission pseudo-differential operator T ∈ L1,0 property with respect to the boundary ∂D, and t(x, y) ≥ 0 off the diagonal {(x, x) : x ∈ RN } in RN × RN . (9) The function τ (x, y) is a local unity function on D; more precisely, τ (x, y) is a smooth function on D ×D, with compact support in a neighborhood of the diagonal ∆∂D , such that, at each point x′ of ∂D, τ (x′ , y) = 1 for y in a neighborhood of x′ in D. The function τ (x, y) depends on the shape of the boundary ∂D. (10) The operator Γr is a boundary condition of order 2 − κ1 , and satisfies the condition Z ′ ′ ′ Q1(x ) + Γr 1(x ) = γ(x ) + r(x′ , y ′ ) [1 − τ (x′ , y ′ )] dy ′ ≤ 0 on ∂D. ∂D

The intuitive meaning of condition (10) is that the jump phenomenon from a point x′ ∈ ∂D to the outside of a neighborhood of x′ on the boundary ∂D is “dominated” by the absorption phenomenon at x′ . In particular, if γ(x′ ) ≡ 0 on ∂D, then condition (10) implies that any Markovian particle does not move by jumps from x′ ∈ ∂D to the outside of a neighborhood V (x′ ) of x′ on the boundary ∂D, since we have the assertion Z r(x′ , y ′ ) [1 − τ (x′ , y ′ )] dy ′ = 0, ∂D

and so, by conditions (7) and (9), r(x′ , y ′ ) = 0 for all y ′ ∈ ∂D \ V (x′ ). It should be noticed that the integral operator Γr u(x′ ) Z N−1 X ∂u ′ ′ (x ) dy (yj − xj ) = r(x′ , y ′ ) u(y ′ ) − τ (x′ , y ′ ) u(x′ ) + ∂xj ∂D j=1 Z + t(x′ , y) [u(y) − u(x′ )] dy, x′ ∈ ∂D, D

2−κ1 1−κ2 is a “regularization” of R ∈ L1,0 (∂D) and T ∈ L1,0 (RN ), since the integrals N−1 X ′ ∂u ′ ′ ′ ′ ′ (yj − xj ) Rr u(x ) = r(x , y ) u(y ) − τ (x , y ) u(x ) + (x ) dy , ∂xj ∂D j=1 ′

Z

′

′

FELLER SEMIGROUPS AND MARKOV PROCESSES ′

Tr u(x ) =

Z

D

7

t(x′ , y) [u(y) − u(x′ )] dy

are both absolutely convergent. Indeed, it suffices to note that the kernels r(x′ , y ′ ) of R and t(x′ , y) of T satisfiy respectively the estimates C′

, x′ , y ′ ∈ ∂D, x′ 6= y ′ , |x′ − y ′ |(N−1)+2−κ1 C ′′ ′ , x′ ∈ ∂D, y ∈ D, 0 ≤ t(x , y) ≤ ′ N+1−κ 2 |x − y|

0 ≤ r(x′ , y ′ ) ≤

where |x′ − y ′ | denotes the geodesic distance between x′ and y ′ with respect to the Riemannian metric of the manifold ∂D. The boundary condition L is called a second-order Ventcel’ boundary condition (cf. [We]). The six terms of L N−1 X

i,j=1

N−1 X ∂ 2u ∂u ′ ′ α (x ) (x ) + β i (x′ ) (x ), ∂xi ∂xj ∂xi ij

′

i=1

γ(x′ )u(x′ ),

∂u µ(x′ ) (x′ ), ∂n

δ(x′ )W u(x′ ), N−1 X

∂u ′ ′ (x ) dy , ∂xj ∂D j=1 Z N−1 X ∂u ′ (x ) dy t(x′ , y) u(y) − u(x′ ) − (yj − xj ) ∂xj D j=1 Z

r(x′ , y ′ ) u(y ′ ) − u(x′ ) −

(yj − xj )

are supposed to correspond to the diffusion along the boundary, the absorption phenomenon, the reflection phenomenon, the viscosity phenomenon and the jump phenomenon on the boundary and the inward jump phenomenon from the boundary, respectively (see Figures 0.3 through 0.5 below). This paper is devoted to the functional analytic approach to the problem of construction of Feller semigroups with Ventcel’ boundary conditions. More precisely, we consider the following problem: Problem. Conversely, given analytic data (W, L), can we construct a Feller semigroup {Tt }t≥0 whose infinitesimal generator is characterized by (W, L) ?

... ... D ... .. .. ..... ... ..... ... . ... . . . . ... ... .. ... . ... . . . . ... ....... ....... ...... .. .. . . ..... . . . . . ∂D ....... ...... .................... .... . . . ....... ............ . ................................................

... ... D ... ... ... .. ..... .... ... . ..... ....... . .... ... ... ... .. .. .. ... ... . ... .. . . . . . . .. ...... ... .. ... . ∂D ....... ...................... ... . . . . . . . ...... ............ ................ ..... ....... ........... ...... ...... .............................................

absorption

reflection Figure 0.3

8

KAZUAKI TAIRA

... ... ... ... ... D ... ... . . ... .. ... . .. .. . ∂D ....... . ...... .... . . ....... . . . ................................................

... ... ... ... ... .. D ... . . . ... . ... .. . .... .... . . . ..... . ∂D ..... ... .. ...... ... ...... ..... ... . . .............. ................ ........ .............. ....... .................. ........................................................

diffusion along the boundary

viscosity

Figure 0.4

... ... ... ... ... .. D . ... . . ... . . . ....... ... .. ... .. . . .... . . . ∂D ..... ....... ... . ...... .... . . .. ......... ....................................

... ... ... .. ... .. D . ... . . ... . . ... .. .... . ∂D ..... .. . . ...... . . . ................................................................... ..............................

jump into the interior

jump on the boundary Figure 0.5

We shall only restrict ourselves to some aspects which have been discussed in our papers [Ta1] through [Ta4]. Our approach is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. It focuses on the relationship between two interrelated subjects in analysis; Feller semigroups and elliptic boundary value problems, providing powerful methods for future research. Now we say that the boundary condition L is transversal on the boundary ∂D if it satisfies the condition Z (0.3) t(x′ , y)dy = +∞ if µ(x′ ) = δ(x′ ) = 0. D

The intuitive meaning of condition (0.3) is that a Markovian particle jumps away “instantaneously” from the points x′ ∈ ∂D where neither reflection nor viscosity phenomenon occurs (which is similar to the reflection phenomenon). Probabilistically, this means that every Markov process on the boundary ∂D is the “trace” on ∂D of trajectories of some Markov process on the closure D = D ∪ ∂D. The next theorem asserts that there exists a Feller semigroup on D corresponding to such a diffusion phenomenon that one of the reflection phenomenon, the viscosity phenomenon and the inward jump phenomenon from the boundary occurs at each point of the boundary ∂D (cf. [Ta4, Theorem 1]): Theorem 1. We define a linear operator A from the space C(D) into itself as follows:

FELLER SEMIGROUPS AND MARKOV PROCESSES

9

(a) The domain of definition D(A) of A is the set (0.4) D(A) = u ∈ C(D) : W u ∈ C(D), Lu = 0 .

(b) Au = W u for all u ∈ D(A). Here W u and Lu are taken in the sense of distributions. Assume that the boundary condition L is transversal on the boundary ∂D. Then the operator A generates a Feller semigroup {Tt }t≥0 on D. The situation may be represented schematically by Figure 0.6 below.

............ ....................... ................................ . . . . . . . . ....... ∂D...... ...... ... . . . .... ..... ... .. ... .. ... .. .. ... D .. ... ... . . ... ..... .. ... ...... .. . . . . . . . . . . . . ......... .................................... ........ ... .... .. ... .......... .......... .. ........ ........ .. . . . . . ....... ..... ... .... .... ... .... ......... ......... ........ ... ... ..... . . . . . . . ......... . ......................... ........................................... Figure 0.6 It should be noticed that Theorem 1 was proved before by Taira [Ta1, Theorem 10.1.3] under some additional conditions, and also by Cancelier [Ca, Th´eor`eme 3.2]. On the other hand, Takanobu and Watanabe [TW] proved a probabilistic version of Theorem 1 in the case where the domain D is the half space RN + (see [TW, Corollary]). Next we generalize Theorem 1 to the non-transversal case. To do this, we assume that (H) There exists a second-order Ventcel’ boundary condition Lν such that Lu = m(x′ ) Lν u + γ(x′ ) u on ∂D, where (3′ ) m ∈ C ∞ (∂D) and m(x′ ) ≥ 0 on ∂D, and the boundary condition Lν is given in local coordinates (x1 , x2 , . . . , xN−1 ) by the formula Lν u(x′ ) ∂u = Qu(x′ ) + µ(x′ ) (x′ ) − δ(x′ )W u(x′ ) + Γu(x′ ) ∂n N−1 N−1 2 X X ∂ u ∂u ′ i αij (x′ ) β (x′ ) := (x′ ) + (x ) ∂x ∂x ∂x i j i i,j=1 i=1

10

KAZUAKI TAIRA

∂u + µ(x′ ) (x′ ) − δ(x′ ) W u(x′ ) ∂n Z N−1 X ∂u ′ ′ r(x′ , y ′ ) u(y ′ ) − u(x′ ) − (yj − xj ) + (x ) dy ∂xj ∂D j=1 Z N−1 X ∂u ′ t(x′ , y) u(y) − u(x′ ) − (yj − xj ) (x ) dy , + ∂x j D j=1

and satisfies the transversality condition Z ′ t(x′ , y) dy = +∞ if µ(x′ ) = δ(x′ ) = 0. (0.3 ) D

We let

′

′

′

M = {x ∈ ∂D : µ(x ) = δ(x ) = 0, Then, by condition (0.3′ ) it follows that

Z

D

t(x′ , y) dy < ∞}.

M = {x′ ∈ ∂D : m(x′ ) = 0},

since we have µ(x′ ) = m(x′ ) µ(x′ ), δ(x′ ) = m(x′ ) δ(x′ ), and t(x′ , y) = m(x′ ) t(x′ , y). Hence we find that the boundary condition L is not transversal on ∂D. Furthermore, we assume that (A) m(x′ ) − γ(x′ ) > 0 on ∂D. The intuitive meaning of conditions (H) and (A) is that a Markovian particle does not stay on ∂D for any period of time until it “dies” at the time when it reaches the set M where the particle is definitely absorbed. Now we introduce a subspace of C(D) which is associated with the boundary condition L. By condition (A), we find that the boundary condition Lu = m(x′ ) Lν u + γ(x′ ) u = 0 on ∂D includes the condition u = 0 on M . With this fact in mind, we let C0 (D \ M ) = {u ∈ C(D) : u = 0 on M }.

The space C0 (D \ M ) is a closed subspace of C(D); hence it is a Banach space. A strongly continuous semigroup {Tt }t≥0 on the space C0 (D \ M ) is called a Feller semigroup on D \ M if it is non-negative and contractive on C0 (D \ M ): f ∈ C0 (D \ M ), 0 ≤ f ≤ 1 on D \ M =⇒ 0 ≤ Tt f ≤ 1 on D \ M .

We define a linear operator W from C0 (D \ M ) into itself as follows: (a) The domain of definition D(W) of W is the set (0.5) D(W) = u ∈ C0 (D \ M ) : W u ∈ C0 (D \ M ), Lu = 0 .

(b) Wu = W u for all u ∈ D(W). The next theorem is a generalization of Theorem 1 to the non-transversal case (cf. [Ta4, Theorem 2]):

FELLER SEMIGROUPS AND MARKOV PROCESSES

11

Theorem 2. Assume that conditions (A) and (H) are satisfied. Then the operator W defined by formula (0.5) generates a Feller semigroup {Tt }t≥0 on D \ M . If Tt is a Feller semigroup on D \M , then there exists a unique Markov transition function pt on D \ M such that Tt f (x) =

Z

pt (x, dy)f (y),

D\M

f ∈ C0 (D \ M ),

and further that pt is the transition function of some strong Markov process. On the other hand, the intuitive meaning of conditions (A) and (H) is that the absorption phenomenon occurs at each point of the set M = {x′ ∈ ∂D : m(x′ ) = 0}. Therefore, Theorem 2 asserts that there exists a Feller semigroup on D \ M corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space D \ M until it “dies” at the time when it reaches the set M . The situation may be represented schematically by Figure 0.7. ...................................................................... ............... ............ ........... ......... . . . . . . . ....... ∂D .......... ..... .. . ..... . . . . . .... . . . ... . . ... . . . .. ... ........... . . . .... . ... .. . . . . . .. . ... . .. . . . . D .. ...... . . ... ... ... . ... ..... .. . . . ... .. . .. ..... ....... .. .. ...... .............. ... ...... ... . . ... .................................. . . ... ........ ..... ....... ..... ...... ....... ..... ........ . . . . . . . . . . . . .......... ....... ..... ............. ..... .......... ....... .......

...... . . . . . . . . . . . . . . . ................................. M

Figure 0.7 It should be noticed that Taira [Ta2] has proved Theorem 2 under the condition that Lν = ∂/∂n and δ(x′ ) ≡ 0 on ∂D, by using the Lp theory of pseudo-differential operators (see [Ta2, Theorem 4]). Finally, we consider the case where all the operators S, T and R are pseudodifferential operators of order less than one. Then we can take σ(x, y) ≡ 1 on D × D, and write the operator W in the following form: (0.1′ )

W u(x) = P u(x) + Sr u(x) X N N X ∂ 2u ∂u ij i := a (x) b (x) (x) + (x) + c(x)u(x) ∂xi ∂xj ∂xi i,j=1 i=1 Z + s(x, y)[u(y) − u(x)]dy , D

where: (4′ ) The integral kernel s(x, y) is the distribution kernel of a properly supported, N pseudo-differential operator S ∈ L1−κ 1,0 (R ), κ > 0, which has the transmission

12

KAZUAKI TAIRA

property with respect to the boundary ∂D, and s(x, y) ≥ 0 off the diagonal {(x, x) : x ∈ RN } in RN × RN . (6′ ) W 1(x) = c(x) ≤ 0 in D. Similarly, the boundary condition L can be written in the following form: ∂u ′ (x ) − δ(x′ )W u(x′ ) + Γu(x′ ) ∂n N−1 N−1 X X ∂ 2u ij ′ i ′ ∂u ′ ′ ′ ′ α (x ) β (x ) := (x ) + (x ) + γ(x )u(x ) ∂x ∂x ∂x i j i i,j=1 i=1

(0.2′ ) Lu(x′ ) = Qu(x′ ) + µ(x′ )

N−1 X ∂u ′ ∂u ′ ′ ′ ′ ′ + µ(x ) (x ) − δ(x )W u(x ) + η(x )u(x ) + ζ i (x′ ) (x ) ∂n ∂x i i=1 Z Z ′ ′ ′ ′ ′ ′ ′ + r(x , y )[u(y ) − u(x )]dy + t(x , y)[u(y) − u(x )]dy , ′

∂D

D

where: (6′ ) The integral kernel r(x′ , y ′ ) is the distribution kernel of a pseudo-differential 1−κ1 operator R ∈ L1,0 (∂D), κ1 > 0, and r(x′ , y ′ ) ≥ 0 off the diagonal {(x′ , x′ ) : x′ ∈ ∂D} in ∂D × ∂D. (7′ ) The integral kernel t(x, y) is the distribution kernel of a properly supported, 1−κ2 pseudo-differential operator T ∈ L1,0 (RN ), κ2 > 0, which has the transmission property with respect to the boundary ∂D, and t(x, y) ≥ 0 off the diagonal {(x, x) : x ∈ RN } in RN × RN . (9′ ) Γ1(x′ ) = η(x′ ) ≤ 0 on ∂D. Then Theorems 1 and 2 may be simplified as follows:

Theorem 3. Assume that the operator W and the boundary condition L are of the forms (0.1′ ) and (0.2′ ), respectively. If the boundary condition L is transversal on the boundary ∂D, then the operator W defined by formula (0.4) generates a Feller semigroup {Tt }t≥0 on D. Theorem 4. Assume that the operator W and the boundary condition L are of the forms (0.1′ ) and (0.2′ ), respectively. If conditions (A) and (H) are satisfied, then the operator W defined by formula (0.6) generates a Feller semigroup {Tt }t≥0 on D \ M. Theorems 1, 2, 3 and 4 solve from the viewpoint of functional analysis the problem of construction of Feller semigroups with Ventcel’ boundary conditions for elliptic Waldenfels operators. The rest of this paper is organized as follows. In Section 1 we present a brief description of basic definitions and results about a class of semigroups (Feller semigroups) associated with Markov processes, which forms a functional analytic background for the proof of Theorems 1 and 2. Section 2 provides a review of basic concepts and results of the theory of pseudodifferential operators – a modern theory of potentials – which will be used in the subsequent sections. In particular we introduce the notion of transmission property due to Boutet de Monvel [Bo], which is a condition about symbols in the normal

FELLER SEMIGROUPS AND MARKOV PROCESSES

13

direction at the boundary. Furthermore, we prove an existence and uniqueness theorem for a class of pseudo-differential operators which enters naturally in the construction of Feller semigroups. Section 3 is devoted to the proof of Theorem 1. We reduce the problem of construction of Feller semigroups to the problem of unique solvability for the boundary value problem (α − W )u = f in D, (λ − L)u = ϕ

on ∂D,

v=0

on ∂D.

and then prove existence theorems for Feller semigroups. Here α is a positive number and λ is a non-negative number. The idea of our approach is stated as follows (cf. [BCP], [RS], [Ta1]). First, we consider the following Dirichlet problem: (α − W )v = f in D, The existence and uniqueness theorem for this problem is well established in the framework of H¨older spaces. We let v = G0α f. The operator G0α is the Green operator for the Dirichlet problem. Then it follows that a function u is a solution of the problem (α − W )u = f in D, (∗) Lu = 0 on ∂D if and only if the function w = u − v is a solution of the problem (α − W )w = 0 in D, Lw = −Lv = −LG0α f

on ∂D.

However, we know that every solution w of the equation (α − W )w = 0 in D can be expressed by means of a single layer potential as follows: w = Hα ψ. The operator Hα is the harmonic operator for the Dirichlet problem. Thus, by using the Green and harmonic operators we can reduce the study of problem (∗) to that of the equation LHα ψ = −LG0α f on ∂D. This is a generalization of the classical Fredholm integral equation. It is known (see [Bo], [Ho], [RS]) that the operator LHα is a pseudo-differential operator of second order on the boundary ∂D.

14

KAZUAKI TAIRA

By using the H¨older space theory of pseudo-differential operators, we can show that if the boundary condition L is transversal on the boundary ∂D, then the operator LHα is bijective in the framework of H¨older spaces. The crucial point in the proof is that we consider the term δ(x′ )(W u|∂D ) of viscosity in the boundary condition Lu = L0 u − δ(x′ )(W u|∂D ) as a term of “perturbation” of the boundary condition L0 u. Therefore, we find that a unique solution u of problem (∗) can be expressed in the form u = G0α f − Hα LHα−1 LG0α f . This formula allows us to verify all the conditions of the generation theorems of Feller semigroups discussed in Subsection 1.2. Intuitively this formula tells us that if the boundary condition L is transversal on the boundary ∂D, then we can “piece together” a Markov process on the boundary ∂D with W -diffusion in the interior D to construct a Markov process on the closure D = D ∪ ∂D. The situation may be represented schematically by Figure 0.8.

....................................... .................. ................ . . . . . . . ........ .. ............. . ..... . . . . . . . . . . ..... ....... .................. . . ............ ... . . .... ... ... ..... ∂D ..... . . . ............ . . . .... ... ..... .. ..... ... . .. . . . .. . .. ........ ...... ... ..... ...... . ... . ... . . .. . ..... . . . . . . . ..... ..... .... ... . . ....... . . . ................................................ ................... .... ....... .................. ... .. . .......... ....... ... ... ......... . . .. . . . . ..... ..... . D ......................... ... .......... . . . ....... ..... ........ ..... ... ..... ......... ..... .. ... ......... . . . ... . . . . . .. ....... . .......... ... . . . . . . ....... ... ... .... .................. ........ ........... ... .... .. .. ..... ......... ..... . .... .... . . . . . ..... .... ...... .... ......... ... ..... ..... . ....... . ..... .................... . . . . . . . ..... . ......... . ...... . .............................................. . . . . . . . . ..................................... Figure 0.8 It seems that our method of construction of Feller semigroups is, in spirit, not far removed from the probabilistic method of construction of diffusion processes by means of Poisson point processes of Brownian excursions used by Watanabe [Wa]. In Section 4 we prove Theorem 2. We explain the idea of the proof. First, we remark that if condition (H) is satisfied, then the boundary condition L can be written in the form Lu = m(x′ ) Lν u + γ(x′ ) u on ∂D, where the boundary condition Lν is transversal on ∂D. Hence, by applying Theorem 1 to the boundary condition Lν we can solve uniquely the following boundary value problem: (α − W )v = f in D, Lν v = 0

on ∂D.

FELLER SEMIGROUPS AND MARKOV PROCESSES

15

We let v = Gνα f. The operator Gνα is the Green operator for the boundary condition Lν . Then it follows that a function u is a solution of the problem (α − W )u = f in D, (∗∗) ′ ′ Lu = m(x ) Lν u + γ(x ) u = 0 on ∂D if and only if the function w =u−v

is a solution of the problem (α − W )w = 0 Lw = −Lv = −γ(x′ ) v

in D, on ∂D.

Thus, as in the proof of Theorem 1 we can reduce the study of problem (∗∗) to that of the equation LHα ψ = −LGνα f = −γ(x′ ) Gνα f

on ∂D.

By using the H¨older space theory of pseudo-differential operators as in the proof of Theorem 1, we can show that if condition (A) is satisfied, then the operator LHα is bijective in the framework of H¨older spaces. Therefore, we find that a unique solution u of problem (∗∗) can be expressed as follows: u = Gνα f − Hα LHα−1 (LGνα f ) .

This formula allows us to verify all the conditions of the generation theorems of Feller semigroups, especially the density of the domain D(W) in C0 (D \ M ). It is worth while pointing out that if we use instead of Gνα the Green operator 0 Gα for the Dirichlet problem as in the proof of Theorem 1, our proof would break down. We do not prove Theorems 3 and 4, since their proofs are essentially the same as those of Theorems 1 and 2, respectively. The content of this paper may be summarized in the following diagram:

Probability

Functional Analysis

Partial Differentital Equations

Markov process (Xt )

FellerR semigroup Tt f (·) = D pt (·, dy)f (y)

Infinitesimal generator Tt = exp[tA]

Semigroup property

Waldenfels operator Ventcel’ boundary condition

pt (x, dy) Markov transition function Markov property

16

KAZUAKI TAIRA

1. Theory of Feller Semigroups This section provides a brief description of basic definitions and results about a class of semigroups (Feller semigroups) associated with Markov processes, which forms a functional analytic background for the proof of Theorems 1 and 2. The results discussed here are adapted from Taira [Ta1, Chapter 9]. 1.1 Markov Processes. In 1828 the English botanist R. Brown observed that pollen grains suspended in water move chaotically, incessantly changing their direction of motion. The physical explanation of this phenomenon is that a single grain suffers innumerable collisions with the randomly moving molecules of the surrounding water. A mathematical theory for Brownian motion was put forward by A. Einstein in 1905. Let p(t, x, y) be the probability density function that a one-dimensional Brownian particle starting at position x will be found at position y at time t. Einstein derived the following formula from statistical mechanical considerations: (y − x)2 1 exp − . p(t, x, y) = √ 2Dt 2πDt Here D is a positive constant determined by the radius of the particle, the interaction of the particle with surrounding molecules, temperature and the Boltzmann constant. This gives an accurate method of measuring Avogadro’s number by observing particles. Einstein’s theory was experimentally tested by J. Perrin between 1906 and 1909. Brownian motion was put on a firm mathematical foundation for the first time by N. Wiener in 1923. Let Ω be the space of continuous functions ω : [0, ∞) 7→ R with coordinates xt (ω) = ω(t) and let F be the smallest σ-algebra in Ω which contains all sets of the form {ω ∈ Ω : a ≤ xt (ω) < b}, t ≥ 0, a < b. Wiener constructed probability measures Px , x ∈ R, on F for which the following formula holds: Px {ω ∈ Ω : a1 ≤ xt1 (ω) < b1 , a2 ≤ xt2 (ω) < b2 , . . . , an ≤ xtn (ω) < bn } Z bn Z b1 Z b2 p(t1 , x, y1 )p(t2 − t1 , y1 , y2 ) . . . ... = a1

a2

an

p(tn − tn−1 , yn−1 , yn ) dy1 dy2 . . . dyn , 0 < t1 < t2 < . . . < tn < ∞.

This formula expresses the “starting afresh” property of Brownian motion that if a Brownian particle reaches a position, then it behaves subsequently as though that position had been its initial position. The measure Px is called the Wiener measure starting at x. Markov processes are an abstraction of the idea of Brownian motion. Let K be a locally compact, separable metric space and B the σ-algebra of all Borel sets in K, that is, the smallest σ-algebra containing all open sets in K. Let (Ω, F , P ) be a probability space. A function X defined on Ω taking values in K is called a random variable if it satisfies the condition {X ∈ E} = X −1 (E) ∈ F

for all E ∈ B.

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17

We express this by saying that X is F /B-measurable. A family {xt }t≥0 of random variables is called a stochastic process, and it may be thought of as the motion in time of a physical particle. The space K is called the state space and Ω the sample space. For a fixed ω ∈ Ω, the function xt (ω), t ≥ 0, defines in the state space K a trajectory or path of the process corresponding to the sample point ω. In this generality the notion of a stochastic process is of course not so interesting. The most important class of stochastic processes is the class of Markov processes which is characterized by the Markov property. Intuitively, this is the principle of the lack of any “memory” in the system. More precisely, (temporally homogeneous) Markov property is that the prediction of subsequent motion of a particle, knowing its position at time t, depends neither on the value of t nor on what has been observed during the time interval [0, t); that is, a particle “starts afresh”. Now we introduce a class of Markov processes which we will deal with in this book. Assume that we are given the following: (1) A locally compact, separable metric space K and the σ-algebra B of all Borel sets in K. A point ∂ is adjoined to K as the point at infinity if K is not compact, and as an isolated point if K is compact. We let K∂ = K ∪ {∂}, B∂ = the σ-algebra in K∂ generated by B. (2) The space Ω of all mappings ω : [0, ∞] → K∂ such that ω(∞) = ∂ and that if ω(t) = ∂ then ω(s) = ∂ for all s ≥ t. Let ω∂ be the constant map ω∂ (t) = ∂ for all t ∈ [0, ∞]. (3) For each t ∈ [0, ∞], the coordinate map xt defined by xt (ω) = ω(t), ω ∈ Ω. (4) For each t ∈ [0, ∞], a mapping ϕt : Ω → Ω defined by ϕt ω(s) = ω(t + s), ω ∈ Ω. Note that ϕ∞ ω = ω∂ and xt ◦ ϕs = xt+s for all t, s ∈ [0, ∞]. (5) A σ-algebra F in Ω and an increasing family {Ft }0≤t≤∞ of sub-σ-algebras of F . (6) For each x ∈ K∂ , a probability measure Px on (Ω, F ). We say that these elements define a (temporally homogeneous) Markov process X = (xt , F , Ft, Px ) if the following conditions are satisfied: (i) For each 0 ≤ t < ∞, the function xt is Ft /B∂ - measurable, that is, {xt ∈ E} ∈ Ft

for all E ∈ B∂ .

(ii) For each 0 ≤ t < ∞ and E ∈ B, the function (1.1)

pt (x, E) = Px {xt ∈ E}

is a Borel measurable function of x ∈ K. (iii) Px {ω ∈ Ω : x0 (ω) = x} = 1 for each x ∈ K∂ . (iv) For all t, h ∈ [0, ∞], x ∈ K∂ and E ∈ B∂ , we have Px {xt+h ∈ E | Ft } = ph (xt , E) a. e.,

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KAZUAKI TAIRA

or equivalently Px (A ∩ {xt+h ∈ E}) =

Z

ph (xt (ω), E) dPx(ω),

A

A ∈ Ft .

Here is an intuitive way of thinking about the above definition of a Markov process. The sub-σ-algebra Ft may be interpreted as the collection of events which are observed during the time interval [0, t]. The value Px (A), A ∈ F , may be interpreted as the probability of the event A under the condition that a particle starts at position x; hence the value pt (x, E) expresses the transition probability that a particle starting at position x will be found in the set E at time t (see Figure 1.1 below). The function pt is called the transition function of the process X . The transition function pt specifies the probability structure of the process. The intuitive meaning of the crucial condition (iv) is that the future behavior of a particle, knowing its history up to time t, is the same as the behavior of a particle starting at xt (ω), that is, a particle starts afresh. A particle moves in the space K until it “dies” at the time when it reaches the point ∂; hence the point ∂ is called the terminal point. With this interpretation in mind, we let ζ(ω) = inf{t ∈ [0, ∞] : xt (ω) = ∂}. The random variable ζ is called the lifetime of the process X . E

........................................ ...... ....... ... ...... .......... . . . . . .................................... ...

..... ... ... ........ ... ... ........ ... ... .......... ... ....... ... ....... ... ....... ..... ....... .. ... ......... . ... ... .................. . ................ ... ................... ... ... ... .... ... .. ....... . . . . . . . . ...... .. . . . . . . . . . . . . . . . ...........

t

• x

Figure 1.1

1.2 Markov Transition Functions and Feller Semigroups. From the point of view of analysis, the transition function is something more convenient than the Markov process itself. In fact, it can be shown that the transition functions of Markov processes generate solutions of certain parabolic partial differential equations such as the classical diffusion equation; and, conversely, these differential equations can be used to construct and study the transition functions and the Markov processes themselves. Let (K, ρ) be a locally compact, separable metric space and B the σ-algebra of all Borel sets in K.

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19

A function pt (x, E), defined for all t ≥ 0, x ∈ K and E ∈ B, is called a (temporally homogeneous) Markov transition function on K if it satisfies the following four conditions: (a) pt (x, ·) is a non-negative measure on B and pt (x, K) ≤ 1 for each t ≥ 0 and each x ∈ K. (b) pt (·, E) is a Borel measurable function for each t ≥ 0 and each E ∈ B. (c) p0 (x, {x}) = 1 for each x ∈ K. (d) (The Chapman-Kolmogorov equation) For any t, s ≥ 0, any x ∈ K and any E ∈ B, we have Z (1.2) pt+s (x, E) = pt (x, dy)ps (y, E). K

Here is an intuitive way of thinking about the above definition of a Markov transition function. The value pt (x, E) expresses the transition probability that a physical particle starting at position x will be found in the set E at time t (see Figure 1.1). Equation (1.2) expresses the idea that a transition from the position x to the set E in time t + s is composed of a transition from x to some position y in time t, followed by a transition from y to the set E in the remaining time s; the latter transition has probability ps (y, E) which depends only on y (see Figure 1.2). Thus a particle “starts afresh”; this property is called the Markov property.

t+s

t

0

. ..... ......... ... ... .. ............................................................................................................................................................................................................................................... ....... ..... .. ... ...... ... ....... .... ..... ........ .. ...... .... ... .... ...... ...... ... .... .... ...... . . . . ... . . . . . . . . . .. . ..... .... .. .... ..... ... .. .... ... ..... . . . . . . . . . ... . . ..... ... . .. . . . . .. ......... . .... .. ... ... .... .. .. .... ... .... . . . . ... .... . . ... .. ... ... .... ..... ... .... ..... .... .. ... .... . .............................................................................................................................................................................................................. .. . ... . . ...... ... ..... .. ..... ... ...... .. . ... . . .. ... ... .. . ... ... .. .... .. ... .. .. ..... ..... ... . ... . .... . ... . ... . ... .... .... .. .. .. ..... ..... ... .. ... .. ..... ..... .... ... .... ..... .... ...... . . . . . . .... . . . . .. ..... .. ... .. . . . . . . . ..... ... .. . . ... ..... ..... .... .... ... ...... ..... . . ..... ... . . . . . . . . .. ..... .... .... .... .... .. ........... ...... .................. . . . . . ........... . ... . . .. ........ ........... ..................................................................................................................................................................................................................................................... . ... . . ... .. ... ... ... ... .... .... . .... . .... .... .... .... ..... .... ...... ..... . . . . . ......... .. ..... ............

..........E .....................

•y

• x

K

Figure 1.2 The Chapman–Kolmogorov equation (1.2) tells us that pt (x, K) is monotonically increasing as t ↓ 0, so that the limit p+0 (x, K) = limt↓0 pt (x, K) exists. A transition function pt is said to be normal if it satisfies the condition p+0 (x, K) = 1 for all x ∈ K. It is known that, for every Markov process, the function pt , defined by formula (1.1), is a transition function. Conversely, every normal transition function corresponds to some Markov process.

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KAZUAKI TAIRA

We add a point ∂ to K as the point at infinity if K is not compact, and as an isolated point if K is compact; so the space K∂ = K ∪ {∂} is compact. Let C(K) be the space of real-valued, bounded continuous functions on K. The space C(K) is a Banach space with the supremum norm kf k∞ = sup |f (x)|. x∈K

We say that a function f ∈ C(K) converges to 0 as x → ∂ if, for each ε > 0, there exists a compact subset E of K such that |f (x)| < ε,

x ∈ K \ E,

and we then write limx→∂ f (x) = 0. We let C0 (K) = f ∈ C(K) : lim f (x) = 0 . x→∂

The space C0 (K) is a closed subspace of C(K); hence it is a Banach space. Note that C0 (K) may be identified with C(K) if K is compact. If we introduce a useful convention Any real-valued function f on K is extended to the space K∂ = K ∪ {∂} by setting f (∂) = 0,

then the space C0 (K) may be identified with the subspace of C(K∂ ) which consists of all functions f satisfying f (∂) = 0: C0 (K) = {f ∈ C(K∂ ) : f (∂) = 0} . Moreover, we can extend a Markov transition function pt on K to a Markov transition function p′t on K∂ as follows: ′ pt (x, E) = pt (x, E), x ∈ K, E ∈ B; p′t (x, {∂}) = 1 − pt (x, K), x ∈ K; ′ pt (∂, K) = 0, p′t (∂, {∂}) = 1.

Intuitively, this means that a Markovian particle moves in the space K until it “dies” at the time when it reaches the point ∂; hence the point ∂ is called the terminal point. Now we introduce some conditions on the measures pt (x, ·) related to continuity in x ∈ K, for fixed t ≥ 0. A Markov transition function pt is called a Feller function if the function Z Tt f (x) = pt (x, dy)f (y) K

is a continuous function of x ∈ K whenever f is in C(K), that is, if we have f ∈ C(K) =⇒ Tt f ∈ C(K).

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21

In other words, the Feller property is equivalent to saying that the measures pt (x, ·) depend continuously on x ∈ K in the usual weak topology, for every fixed t ≥ 0. We say that pt is a C0 -function if the space C0 (K) is an invariant subspace of C(K) for the operators Tt : f ∈ C0 (K) =⇒ Tt f ∈ C0 (K). The Feller or C0 -property deals with continuity of a Markov transition function pt (x, E) in x, and does not, by itself, have no concern with continuity in t. We give a necessary and sufficient condition on pt (x, E) in order that its associated operators {Tt }t≥0 be strongly continuous in t on the space C0 (K): lim kTt+s f − Tt f k∞ = 0, s↓0

f ∈ C0 (K).

A Markov transition function pt on K is said to be uniformly stochastically continuous on K if the following condition is satisfied: For each ε > 0 and each compact E ⊂ K, we have lim sup [1 − pt (x, Uε (x))] = 0, t↓0 x∈E

where Uε (x) = {y ∈ K : ρ(x, y) < ε} is an ε-neighborhood of x. Then we have the following (see [Ta1, Theorem 9.2.3]): Theorem 1.1. Let pt be a C0 -transition function on K. Then the associated operators {Tt }t≥0 , defined by the formula Z (1.3) Tt f (x) = pt (x, dy)f (y), f ∈ C0 (K), K

are strongly continuous in t on C0 (K) if and only if pt is uniformly stochastically continuous on K and satisfies the following condition: (L) For each s > 0 and each compact E ⊂ K, we have lim sup pt (x, E) = 0.

x→∂ 0≤t≤s

A family {Tt }t≥0 of bounded linear operators acting on C0 (K) is called a Feller semigroup on K if it satisfies the following three conditions: (i) Tt+s = Tt · Ts , t, s ≥ 0; T0 = I. (ii) The family {Tt } is strongly continuous in t for t ≥ 0: lim kTt+s f − Tt f k∞ = 0, s↓0

f ∈ C0 (K).

(iii) The family {Tt } is non-negative and contractive on C0 (K): f ∈ C0 (K), 0 ≤ f ≤ 1 on K =⇒ 0 ≤ Tt f ≤ 1 on K. The next theorem gives a characterization of Feller semigroups in terms of Markov transition functions (see [Ta1, Theorem 9.2.6]):

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Theorem 1.2. If pt is a uniformly stochastically continuous C0 -transition function on K and satisfies condition (L), then its associated operators {Tt }t≥0 form a Feller semigroup on K. Conversely, if {Tt }t≥0 is a Feller semigroup on K, then there exists a uniformly stochastically continuous C0 -transition pt on K, satisfying condition (L), such that formula (1.3) holds. 1.3 Generation Theorems of Feller Semigroups. If {Tt }t≥0 is a Feller semigroup on K, we define its infinitesimal generator A by the formula (1.4)

Au = lim t↓0

Tt u − u , t

provided that the limit (1.4) exists in the space C0 (K). More precisely, the generator A is a linear operator from the space C0 (K) into itself defined as follows. (1) The domain D(A) of A is the set D(A) = {u ∈ C0 (K) : the limit (1.4) exists} . Tt u − u , u ∈ D(A). t The next theorem is a version of the Hille–Yosida theorem adapted to the present context (see [Ta1, Theorem 9.3.1 and Corollary 9.3.2]): (2) Au = limt↓0

Theorem 1.3. (i) Let {Tt }t≥0 be a Feller semigroup on K and A its infinitesimal generator. Then we have the following: (a) The domain D(A) is everywhere dense in the space C0 (K). (b) For each α > 0, the equation (αI − A)u = f has a unique solution u in D(A) for any f ∈ C0 (K). Hence, for each α > 0, the Green operator (αI − A)−1 : C0 (K) −→ C0 (K) can be defined by the formula u = (αI − A)−1 f,

f ∈ C0 (K).

(c) For each α > 0, the operator (αI − A)−1 is non-negative on the space C0 (K): f ∈ C0 (K), f ≥ 0

on K =⇒ (αI − A)−1 f ≥ 0

on K.

(d) For each α > 0, the operator (αI − A)−1 is bounded on the space C0 (K) with norm 1 k(αI − A)−1 k ≤ . α (ii) Conversely, if A is a linear operator from the space C0 (K) into itself satisfying condition (a) and if there exists a constant α0 ≥ 0 such that, for all α > α0 , conditions (b) through (d) are satisfied, then the operator A is the infinitesimal generator of some Feller semigroup {Tt }t≥0 on K.

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23

Corollary 1.4. Let K be a compact metric space and let A be the infinitesimal generator of a Feller semigroup on K. Assume that the constant function 1 belongs to the domain D(A) of A and that we have, for some constant c, A 1 ≤ −c

on K.

Then the operator A′ = A + cI is the infinitesimal generator of some Feller semigroup on K. Finally, we give useful criteria in terms of the maximum principle (see [Ta1, Theorem 9.3.3 and Corollary 9.3.4]): Theorem 1.5. Let K be a compact metric space. Then we have the following assertions: (i) Let B be a linear operator from the space C(K) = C0 (K) into itself, and assume that (α) The domain D(B) of B is everywhere dense in the space C(K). (β) There exists an open and dense subset K0 of K such that if u ∈ D(B) takes a positive maximum at a point x0 of K0 , then we have Bu(x0 ) ≤ 0. Then the operator B is closable in the space C(K). (ii) Let B be as in part (i), and further assume that (β ′ ) If u ∈ D(B) takes a positive maximum at a point x′ of K, then we have Bu(x′ ) ≤ 0. (γ) For some α0 ≥ 0, the range R(α0 I − B) of α0 I − B is everywhere dense in the space C(K). Then the minimal closed extension B of B is the infinitesimal generator of some Feller semigroup on K. Corollary 1.6. Let A be the infinitesimal generator of a Feller semigroup on a compact metric space K and C a bounded linear operator on C(K) into itself. Assume that either C or A′ = A + C satisfies condition (β ′ ). Then the operator A′ is the infinitesimal generator of some Feller semigroup on K. 2. Theory of Pseudo-Differential Operators In this section we present a brief description of basic concepts and results of the H¨older space theory of pseudo-differential operators which will be used in the subsequent sections. In particular we introduce the notion of transmission property due to Boutet de Monvel [Bo], which is a condition about symbols in the normal direction at the boundary. Furthermore, we prove an existence and uniqueness theorem for a class of pseudo-differential operators which enters naturally in the construction of Feller semigroups. For detailed studies of pseudo-differential operators, the reader is referred to Chazarain–Piriou [CP], H¨ormander [Ho], Kumano-go [Ku], Rempel–Schulze [RS] and Taylor [Ty].

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2.1 Function Spaces. Let Ω be an open subset of Rn . A Lebesgue measurable function u on Ω is said to be essentially bounded if there exists a constant C > 0 such that |u(x)| ≤ C almost everywhere (a.e.) in Ω. We define ess supx∈Ω |u(x)| = inf{C : |u(x)| ≤ C a.e. in Ω}. We let L∞ (Ω) = the space of equivalence classes of essentially bounded, Lebesgue measurable functions on Ω. The space L∞ (Ω) is a Banach space with the norm kuk∞ = ess supx∈Ω |u(x)|. If m is a non-negative integer, we let W m,∞ (Ω) = the space of equivalence classes of functions u ∈ L∞ (Ω) all of whose derivatives ∂ α u,

|α| ≤ m, in the sense of distributions are

in L∞ (Ω).

The space W m,∞ (Ω) is a Banach space with the norm kukm,∞ =

X

|α|≤m

k∂ α uk∞ .

Here and in the following we use the shorthand ∂ , 1 ≤ j ≤ n, ∂xj ∂ α = ∂1α1 ∂2α2 . . . ∂nαn , α = (α1 , α2 , . . . , αn ), ∂j =

for derivatives on Rn . We remark that W 0,∞ (Ω) = L∞ (Ω);

k · k0,∞ = k · k∞ .

Now we let C(Ω) = the space of continuous functions on Ω. If k is a positive integer, we let C k (Ω) = the space of functions of class C k on Ω.

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25

Further we let C(Ω) = the space of functions in C(Ω) having continuous extensions to the closure Ω of Ω. If k is a positive integer, we let C k (Ω) = the space of functions in C k (Ω) all of whose derivatives of order ≤ k have continuous extensions to Ω. If Ω is bounded, then the space C k (Ω) is a Banach space with the norm kukC k (Ω) = sup |∂ α u(x)|. x∈Ω |α|≤k

Let 0 < θ < 1. A function u defined on Ω is said to be H¨ older continuous with exponent θ if the quantity [u]θ;Ω = sup x,y∈Ω x6=y

|u(x) − u(y)| |x − y|θ

is finite. We say that u is locally H¨ older continuous with exponent θ if it is H¨older continuous with exponent θ on compact subsets of Ω. We let C θ (Ω) = the space of functions in C(Ω) which are locally H¨older continuous with exponent θ on Ω. If k is a positive integer, we let C k+θ (Ω) = the space of functions in C k (Ω) all of whose k-th order derivatives are locally H¨older continuous with exponent θ on Ω. Further we let C θ (Ω) = the space of functions in C(Ω) which are H¨older continuous with exponent θ on Ω. If k is a positive integer, we let C k+θ (Ω) = the space of functions in C k (Ω) all of whose k-th order derivatives are H¨older continuous with exponent θ on Ω.

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KAZUAKI TAIRA

If Ω is bounded, then the space C k+θ (Ω) is a Banach space with the norm kukC k+θ (Ω) = kukC k (Ω) + sup [∂ α u]θ;Ω . |α|=k

If M is an n-dimensional compact smooth manifold without boundary and m is a non-negative integer, then the spaces W m,∞ (M ) and C m+θ (M ) are defined respectively to be locally the spaces W m,∞ (Rn ) and C m+θ (Rn ), upon using local coordinate systems flattening out M , together with a partition of unity. The norms of the spaces W m,∞ (M ) and C m+θ (M ) will be denoted by k·km,∞ and k·kC m+θ (M ) , respectively. We recall the following results (see [Tr]): (I) If k is a positive integer, then we have W k,∞ (M ) = {ϕ ∈ C k−1 (M ) : max

|α|≤k−1

sup x,y∈M x6=y

|∂ α ϕ(x) − ∂ α ϕ(y)| < ∞}, |x − y|

where |x − y| is the geodesic distance between x and y with respect to the Riemannian metric of M . (II) The space C k+θ (M ) is a real interpolation space between W k,∞ (M ) and W k+1,∞ (M ); more precisely, we have C k+θ (M ) = W k,∞ (M ), W k+1,∞(M ) θ,∞ K(t, u) k,∞ = u∈W (M ) : sup 0 where K(t, u) =

inf

u=u0 +u1

(ku0 kk,∞ + tku1 kk+1,∞ ) .

2.2 Pseudo-Differential Operators. Let Ω be an open subset of Rn . If m ∈ R and 0 ≤ δ < ρ ≤ 1, we let m Sρ,δ (Ω × RN ) = the set of all functions a ∈ C ∞ (Ω × RN ) with the property

that, for any compact K ⊂ Ω and multi-indices α, β, there exists a constant CK,α,β > 0 such that we have, for all x ∈ K and θ ∈ RN , α β ∂θ ∂x a(x, θ) ≤ CK,α,β (1 + |θ|)m−ρ|α|+δ|β| .

m The elements of Sρ,δ (Ω × RN ) are called symbols of order m. We set \ m S −∞ (Ω × RN ) = Sρ,δ (Ω × RN ). m∈R

For example, if ϕ(ξ) ∈ S(RN ), then it follows that ϕ(ξ) ∈ S −∞ (Ω × RN ). More precisely, we have the following: S −∞ (Ω × RN ) = C ∞ (Ω, S(RN )).

FELLER SEMIGROUPS AND MARKOV PROCESSES

27

m

If aj ∈ Sρ,δj (Ω × RN ) is a sequence of symbols of decreasing order, then there m0 exists a symbol a ∈ Sρ,δ (Ω × RN ), unique modulo S −∞ (Ω × RN ), such that we have, for all k > 0, k−1 X mk a− aj ∈ Sρ,δ (Ω × RN ). j=0

In this case, we write

∞ X

a∼

aj .

j=0

P The formal sum j aj is called an asymptotic expansion of a. m A symbol a(x, θ) ∈ S1,0 (Ω × RN ) is said to be classical if there exist smooth functions aj (x, θ), positively homogeneous of degree m − j in θ for |θ| ≥ 1, such that ∞ X aj . a∼ j=0

The homogeneous function a0 of degree m is called the principal part of a. We let m Scl (Ω × RN ) = the set of all classical symbols of order m. m m Moreover, if Scl (RN ) is the subspace of all x-independent elements of Scl (Ω×RN ), then we have the following: m m (RN )). (Ω × RN ) = C ∞ (Ω, Scl Scl

Let Ω be an open subset of Rn and m ∈ R. A pseudo-differential operator of order m on Ω is a Fourier integral operator of the form ZZ Au(x) = ei(x−y)·ξ a(x, y, ξ)u(y) dydξ , u ∈ C0∞ (Ω), Ω×Rn

m (Ω×Ω×Rn ). Here the integral is taken in the sense of oscillatory with some a ∈ Sρ,δ integrals. We let

Lm ρ,δ (Ω) = the set of all pseudo-differential operators of order m on Ω, and set L−∞ (Ω) =

\

Lm ρ,δ (Ω).

m∈R

Recall that a continuous linear operator A : C0∞ (Ω) −→ D ′ (Ω) is said to be properly supported if the following two conditions are satisfied: (a) For any compact subset K of Ω, there exists a compact subset K ′ of Ω such that supp v ⊂ K =⇒ supp Av ⊂ K ′ .

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(b) For any compact subset K ′ of Ω, there exists a compact subset K ⊃ K ′ of Ω such that supp v ∩ K = ∅ =⇒ supp Av ∩ K ′ = ∅. m If A ∈ Lm ρ,δ (Ω), we can choose a properly supported operator A0 ∈ Lρ,δ (Ω) such that A − A0 ∈ L−∞ (Ω), and define

σ(A) = the equivalence class of the complete symbol of A0 m in the factor class Sρ,δ (Ω × Rn )/S −∞ (Ω × Rn ).

The equivalence class σ(A) does not depend on the operator A0 chosen, and is called the complete symbol of A. We shall often identify the complete symbol σ(A) with a representative in the m class Sρ,δ (Ω × Rn ) for notational convenience, and call any member of σ(A) a complete symbol of A. A pseudo-differential operator A ∈ Lm 1,0 (Ω) is said to be classical if its complete m (Ω × Rn ). symbol σ(A) has a representative in the class Scl We let Lm cl (Ω) = the set of all classical pseudo-differential operators of order m on Ω. If A ∈ Lm cl (Ω), then the principal part of σ(A) has a canonical representative σA (x, ξ) ∈ C ∞ (Ω × (Rn \ {0})) which is positively homogeneous of degree m in the variable ξ. The function σA (x, ξ) is called the homogeneous principal symbol of A. Now we define the concept of a pseudo-differential operator on a manifold, and transfer all the machinery of pseudo-differential operators to manifolds. Let M be an n-dimensional, compact smooth manifold without boundary and 1 − ρ ≤ δ < ρ ≤ 1. A continuous linear operator A : C ∞ (M ) −→ C ∞ (M ) is called a pseudo-differential operator of order m ∈ R if it satisfies the following two conditions: (i) The distribution kernel of A is of class smooth off the diagonal ∆M = {(x, x) : x ∈ M } in M × M . (ii) For any chart (U, χ) on M , the mapping Aχ : C0∞ (χ(U )) −→ C ∞ (χ(U )) u 7−→ A(u ◦ χ) ◦ χ−1

belongs to the class Lm ρ,δ (χ(U )). C0∞ (U ) x ∗ χ

A

−−−−→

C ∞ (U ) χ∗ y

C0∞ (χ(U )) −−−−→ C ∞ (χ(U )) Aχ

FELLER SEMIGROUPS AND MARKOV PROCESSES

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Here χ∗ v = v ◦ χ is the pull-back of v by χ and χ∗ u = u ◦ χ−1 is the push-forward of u by χ, respectively. We let Lm ρ,δ (M ) = the set of all pseudo-differential operators of order m on M , and set L−∞ (M ) =

\

Lm ρ,δ (M ).

m∈R

Some results about pseudo-differential operators on Rn are also true for pseudodifferential operators on M , since pseudo-differential operators on M are defined to be locally pseudo-differential operators on Rn . For example we have the following results: (1) A pseudo-differential operator A extends to a continuous linear operator A : D ′ (M ) −→ D ′ (M ). (2) sing supp Au ⊂ sing supp u , u ∈ D ′ (M ). (3) A continuous linear operator A : C ∞ (M ) −→ D ′ (M ) is a regularizer if and only if it is in L−∞ (M ). (4) The class Lm ρ,δ (M ), 1 − ρ ≤ δ < ρ ≤ 1, is stable under the operations of composition of operators and taking the transpose or adjoint of an operator. (5) A pseudo-differential operator A ∈ Lm 1,0 (M ) extends to a continuous linear k+θ k−m+θ operator A : C (M ) → C (M ) for any integer k ≥ m. A pseudo-differential operator A ∈ Lm 1,0 (M ) is said to be classical if, for any chart (U, χ) on M , the mapping Aχ : C0∞ (χ(U )) −→ C ∞ (χ(U )) belongs to the class Lm cl (χ(U )). We let Lm cl (M ) = the set of all classical pseudo-differential operators of order m on M . We observe that L−∞ (M ) =

\

Lm cl (M ).

m∈R

Let A ∈ Lm cl (M ). If (U, χ) is a chart on M , there is associated a homogeneous principal symbol σAχ ∈ C ∞ (χ(U ) × (Rn \ {0})). Then, by smoothly patching together the functions σAχ we can obtain a smooth function σA (x, ξ) on T ∗ (M ) \ {0} = {(x, ξ) ∈ T ∗ (M ) : ξ 6= 0}, which is positively homogeneous of degree m in the variable ξ. The function σA (x, ξ) is called the homogeneous principal symbol of A. A classical pseudo-differential operator A ∈ Lm cl (M ) is said to be elliptic of order m if its homogeneous principal symbol σA (x, ξ) does not vanish on the bundle T ∗ (M ) \ {0} of non-zero cotangent vectors. Then we have the following: (6) An operator A ∈ Lm cl (M ) is elliptic if and only if there exists a parametrix −m B ∈ Lcl (M ) for A: AB ≡ I mod L−∞ (M ), BA ≡ I

mod L−∞ (M ).

30

KAZUAKI TAIRA

Finally, we introduce the notion of transmission property, due to Boutet de Monvel [Bo], which is a condition about symbols in the normal direction at the boundary. We let m m (Rn+ × Rn+ × Rn ) S1,0 (Rn+ × Rn+ × Rn ) = the space of symbols in S1,0

m which have an extension in S1,0 (Rn × Rn × Rn ).

m We say that a symbol a(x, y, ξ) ∈ S1,0 (Rn+ ×Rn+ ×Rn ) has the transmission property with respect to the boundary Rn−1 if the function a(x, y, ξ) and all its derivatives admit an expansion of the form α β ∂ ∂ a(x, y, ξ) x=y ∂x ∂y xn =0

=

m X j=0

bj (x′ , ξ ′ )ξnj +

∞ X

k

ak (x′ , ξ ′ )

k=−∞

(< ξ ′ > −iξn )

(< ξ ′ > +iξn )

k+1

,

m−j where bj ∈ S1,0 (Rn−1 × Rn−1 ) and the ak form a rapidly decreasing sequence in m+1 S1,0 (Rn−1 × Rn−1 ) with respect to k, and < ξ ′ >= (1 + |ξ ′ |2 )1/2 . We let m n n Lm 1,0 (R+ ) = the space of pseudo-differential operators in L1,0 (R+ ) which can n be extended to a pseudo-differential operator in Lm 1,0 (R ). n A pseudo-differential operator A ∈ Lm 1,0 (R+ ) is said to have the transmission property with respect to the boundary Rn−1 if any complete symbol of A has the transmission property with respect to the boundary Rn−1 . It is known that a n pseudo-differential operator A ∈ Lm 1,0 (R+ ) has the transmission property with respect to the boundary Rn−1 if the restriction of A(u0 ) to Rn+ has a smooth extension to Rn for every u ∈ C0∞ (Rn+ ), where u0 is the extension of u to Rn by zero outside Rn+ .

...................................... . . . . . ..... ..... .... ..... ... .. ... M Ω .. ... .. ... .... .... .... .... .... .... .... . .. ... .... ... .. ........ .... .... . .... . .. .. .. . .... ..... ...... .. ∂Ω ............ . . . . ...... ... ........ . . . . . . ... ... ................ ............................... .... ... . ... .. ... . .... ... . ..... . . ... ....... .................................... Figure 2.1

FELLER SEMIGROUPS AND MARKOV PROCESSES

31

It should be emphasized that the notion of transmission property may be transferred to manifolds with boundary. Indeed, if Ω is a relatively compact open subset of an n-dimensional paracompact smooth manifold M without boundary (see Figure 2.1), then the notion of transmission property can be extended to the class Lm 1,0 (M ), upon using local coordinate systems flattening out the boundary ∂Ω. Then we have the following results (see [Bo], [RS]): (I) If a pseudo-differential operator A ∈ Lm 1,0 (M ) has the transmission property with respect to the boundary ∂Ω, then the operator AΩ : C ∞ (Ω) −→ C ∞ (Ω) u 7−→ A(u0 )|Ω

maps C ∞ (Ω) continuously into itself, where u0 is the extension of u to M by zero outside Ω. (II) If a pseudo-differential operator A ∈ Lm 1,0 (M ) has the transmission property, k+θ then the operator AΩ maps C (Ω) continuously into C k−m+θ (Ω) for any integer k ≥ m and 0 < θ < 1. 2.3 Unique Solvability Theorem for Pseudo-Differential Operators. The next result will play an essential role in the construction of Feller semigroups in Sections 3 and 4 (see [Ta3, Theorem 2.1]): Theorem 2.1. Let T be a classical pseudo-differential operator of second order on an n-dimensional, compact smooth manifold M without boundary such that T = P + S, where: (a) The operator P is a second-order degenerate elliptic differential operator on M with non-positive principal symbol, and P 1 ≤ 0 on M . (b) The operator S is a classical pseudo-differential operator of order 2 − κ, κ > 0, on M and its distribution kernel s(x, y) is non-negative off the diagonal ∆M = {(x, x) : x ∈ M } in M × M . (c) T 1 = P 1 + S1 ≤ 0 on M . Then, for each integer k ≥ 1, there exists a constant λ = λ(k) > 0 such that, for any f ∈ C k+θ (M ), we can find a function ϕ ∈ C k+θ (M ) satisfying the equation (T − λI)ϕ = f

on M ,

and the estimate kϕkC k+θ (M ) ≤ Ck+θ (λ)kf kC k+θ (M ) . Here Ck+θ (λ) > 0 is a constant independent of f . Proof. We prove Theorem 2.1 by using a method of elliptic regularization (see [OR, Chapter I]), just as in the proof of Th´eor`eme 4.5 of Cancelier [Ca]. So we only give a sketch of the proof. (1) We recall the following results:

32

KAZUAKI TAIRA

Theorem 2.2. Let T = P + S be a classical pseudo-differential operator of second order on M as in Theorem 2.1. Assume that T 1 = P 1 + S1 < 0

on M .

Then we have, for all ϕ ∈ C 2 (M ), kϕkC(M ) ≤

kT ϕkC(M ) . − maxM T 1

Theorem 2.2 is a compact manifold version of Theorem A.2 in Appendix. Theorem 2.3. Let T = P + S be a classical pseudo-differential operator of second order on M as in Theorem 2.1. Assume that the operator T is elliptic on M and satisfies the condition T 1 = P 1 + S1 < 0 on M . Then, for each integer k ≥ 0, the operator T : C k+2+θ (M ) −→ C k+θ (M ) is bijective. Since T is elliptic and its principal symbol is real, it follows from an application of Taira [Ta1, Corollary 6.7.12] that ind T = dim N (T ) − codim R(T ) = 0. However, Theorem 2.2 tells us that T is injective, that is, dim N (T ) = 0. Hence we obtain that codim R(T ) = 0, which proves that T is surjective. (2) First, we prove Theorem 2.1 for the space W 1,∞ (M ): Claim I. There exists a constant λ = λ(1) > 0 such that for any f ∈ W 1,∞ (M ) we can find a function ϕ ∈ W 1,∞ (M ) satisfying (T − λI)ϕ = f

on M ,

and kϕk1,∞ ≤ C1 kf k1,∞ . Here C1 > 0 is a constant independent of f . Proof. (2-i) Let {(Uα , χα )}ℓα=1 be a finite open covering of M by local charts, and let {σα }ℓα=1 be a family of functions in C ∞ (M × M ) such that supp σα ⊂ Uα × Uα , and ℓ X

α=1

σα (x, y) = 1 in a neighborhood of the diagonal ∆M = {(x, x) : x ∈ M }.

FELLER SEMIGROUPS AND MARKOV PROCESSES

33

Then the operator T = P + S can be written, in terms of local coordinates (x1 , x2 , . . . , xn ), in the form n X ∂ϕ ∂ 2ϕ β i (x) (x) + (x) + γ(x)ϕ(x) α (x) T ϕ(x) = ∂xi ∂xj ∂xi i=1 i,j=1 Z n X ∂ϕ + s(x, y) ϕ(y) − σ(x, y) ϕ(x) + (yi − xi ) (x) dy. ∂x i M i=1 n X

ij

Here: ij (a) The α are the components of a smooth symmetric contravariant tensor of 2 type 0 on M satisfying n X

i,j=1

ij

α (x)ξi ξj ≥ 0,

x ∈ M, ξ =

n X j=1

ξj dxj ∈ Tx∗ (M ),

where Tx∗ (M ) is the cotangent space of M at x. Pℓ (b) σ(x, y) = α=1 σα (x, y). (c) The density dy isR a strictly positive density on M . (d) T 1(x) = γ(x) + M s(x, y)[1 − σ(x, y)]dy ≤ 0 on M . Furthermore, it should be noticed that there exists a constant C > 0 such that the distribution kernel s(x, y) of S ∈ L2−κ cl (M ), κ > 0, satisfies the estimate C , |x − y|n+2−κ

|s(x, y)| ≤

(x, y) ∈ (M × M ) \ ∆M ,

where |x − y| is the geodesic distance between x and y with respect to the Riemannian metric of M (see [CM, Chapitre IV, Proposition 1]). Hence we find that the integral Z

M

n X ∂ϕ (x) dy s(x, y) ϕ(y) − σ(x, y) ϕ(x) + (yi − xi ) ∂x i i=1

is absolutely convergent, since κ > 0 and σ(x, y) = 1 in a neighborhood of the diagonal ∆M . Now, for each ϕ ∈ C 1 (M ) we introduce a continuous function B(ϕ, ϕ)(x) = 2

n X

αij (x)

i,j=1

+

Z

M

∂ϕ ∂ϕ (x) (x) ∂xi ∂xj 2

s(x, y) (ϕ(y) − ϕ(x)) dy − T 1(x) · ϕ(x)2 ,

x ∈ M.

It should be noticed that the function B(ϕ, ϕ) is non-negative on M for all ϕ ∈ C 1 (M ). The next result may be proved just as in the proof of Th´eor`eme 4.1 of Cancelier [Ca].

34

KAZUAKI TAIRA

Lemma 2.4. Let {Xj }rj=1 be a family of real smooth vector fields on M such that the Xj span the tangent space Tx (M ) at each point x of M . If ϕ ∈ C ∞ (M ), we let p1 (x) =

r X j=1

and R1 (x) = T p1 (x) −

|Xj ϕ(x)|2 ,

r X

x ∈ M,

B(Xj ϕ, Xj ϕ)(x),

j=1

x ∈ M.

Then, for each η > 0, there exist constants β0 > 0 and β1 > 0 such that we have, for all ϕ ∈ C ∞ (M ), (2.1)

|R1 (x)| ≤ η

r X j=1

B(Xj ϕ, Xj ϕ)(x) + β0 kϕk2C(M )

1 + β1 kϕk2C 1 (M ) + kT ϕk2C 1 (M ) , 2

x ∈ M.

Remark 2.5. The constants β0 and β1 are uniform for the operators T + εΛ, 0 ≤ ε ≤ 1, where Λ is a second-order elliptic differential operator on M defined by the formula Λ=− =

r X

Xj∗ Xj

j=1 r X

Xj2

j=1

+

r X j=1

div Xj · Xj .

(2-ii) First, let f be an arbitrary element of C ∞ (M ). Since the operator T + εΛ − λI is elliptic for all ε > 0 and (T + εΛ − λI)1 = T 1 − λ < 0 on M for λ > 0, it follows from an application of Theorem 2.3 that we can find a unique function ϕε ∈ C ∞ (M ) such that (T + εΛ − λI)ϕε = f

on M .

Furthermore, by applying Theorem 2.2 to the operator T + εΛ − λI we obtain that (2.2)

kϕε kC(M ) ≤

1 kf kC(M ) , λ

since (T + εΛ − λI)1 = T 1 − λ ≤ −λ on M . Let x0 be a point of M at which the function p1 (x) =

r X j=1

|Xj ϕ(x)|2

FELLER SEMIGROUPS AND MARKOV PROCESSES

35

attains its positive maximum. Then we have Λp1 (x0 ) =

X r j=1

Xj2

p1 (x0 ) ≤ 0

and also n X

T p1 (x0 ) =

αij (x0 )

i,j=1

∂ 2 p1 (x0 ) + γ(x0 )p1 (x0 ) ∂xi ∂xj

Z

s(x0 , y)[p1 (y) − σ(x0 , y)p1 (x0 )]dy Z ≤ γ(x0 ) + s(x0 , y)[1 − σ(x0 , y)]dy p1 (x0 ) M Z + s(x0 , y)[p1 (y) − p1 (x0 )]dy +

M

M

≤ T 1(x0 ) · p1 (x0 ).

Hence, by using inequality (2.1) with η := 1/2 (and Remark 2.5) and inequality (2.2) we obtain that λp1 (x0 ) ≤ (λ − T 1(x0 ))p1 (x0 ) − εΛp1 (x0 )

≤ (λ − T − εΛ)p1 (x0 ) r X B(Xj ϕε , Xj ϕε )(x0 ) = − (T + εΛ − λ)p1 (x0 ) − j=1

− ≤−

r X

B(Xj ϕε , Xj ϕε )(x0 )

j=1 r X

1 2

j=1

B(Xj ϕε , Xj ϕε )(x0 ) + β0 kϕε k2C(M )

1 + β1 kϕε k2C 1 (M ) + kf k2C 1 (M ) 2 1 β0 ≤ 2 kf k2C(M ) + β1 kϕε k2C 1 (M ) + kf k2C 1 (M ) . λ 2 This proves that (2.3)

(λ − β1 )kϕε k2C 1 (M ) ≤ λ kϕε k2C(M ) + p1 (x0 ) − β1 kϕε k2C 1 (M ) ≤

1 β0 1 kf k2C(M ) + 2 kf k2C(M ) + kf k2C 1 (M ) . λ λ 2

Therefore, if λ > 0 is so large that λ > β1 ,

36

KAZUAKI TAIRA

then it follows from inequality (2.3) that (2.4)

kϕε k2C 1 (M ) ≤ Ckf k2C 1 (M ) ,

where C > 0 is a constant independent of ε > 0. (2-iii) Now let f be an arbitrary element of W 1,∞ (M ). Then we can find a ∞ sequence {fp }∞ p=1 in C (M ) such that fp −→ f in C(M ), kfp kC 1 (M ) ≤ kf k1,∞ . If ϕεp is a unique solution in C ∞ (M ) of the equation (2.5)

(T + εΛ − λI)ϕεp = fp

on M ,

it follows from an application of inequality (2.4) that kϕεp k2C 1 (M ) ≤ Ckfp k2C 1 (M ) ≤ Ckf k21,∞ . This proves that the sequence {ϕεp } is uniformly bounded and equicontinuous. Hence, by virtue of the Ascoli–Arzel`a theorem we can choose a subsequence {ϕε′ p′ } which converges uniformly to a function ϕ ∈ C(M ), as ε′ ↓ 0 and p′ → ∞. Furthermore, since the unit ball in L2 (M ) is sequentially weakly compact (see [Yo, Chapter V, Section 2, Theorem 1]), we may assume that the sequence {∂j ϕε′ p′ } converges weakly to a function ψj in L2 (M ), for each 1 ≤ j ≤ n. Then we have ∂j ϕ = ψj ∈ L2 (M ),

1 ≤ j ≤ n.

On the other hand it is easy to verify that the set √ K = {g ∈ L2 (M ) : kgk∞ ≤ C kf k1,∞ }

is convex and strongly closed in L2 (M ). Thus it follows from an application of Mazur’s theorem (see [Yo, Chapter V, Section 1, Theorem 11]) that the set K is weakly closed in L2 (M ). However, we have ∂j ϕε′ p′ ∈ K, ∂j ϕε′ p′ −→ ψj weakly in L2 (M ) for each 1 ≤ j ≤ n. Hence we find that that is,

∂j ϕ = ψj ∈ K, k∂j ϕk∞ ≤

√

1 ≤ j ≤ n,

C kf k1,∞ ,

Summing up, we have proved that ϕ ∈ W 1,∞ (M ),

1 ≤ j ≤ n.

kϕk1,∞ ≤ C1 kf k1,∞ , where C1 > 0 is a constant independent of f . Finally, by letting ε′ ↓ 0 and p′ → ∞ in the equation

(2.5′ )

we obtain that

(T + ε′ Λ − λI)ϕε′ p′ = fp′ (T − λI)ϕ = f

on M ,

on M .

The proof of Claim I is complete. (3) Similarly we can prove Theorem 2.1 for the spaces W m,∞ (M ) where m ≥ 2:

FELLER SEMIGROUPS AND MARKOV PROCESSES

37

Claim II. For each integer m ≥ 2, there exists a constant λ = λ(m) > 0 such that for any f ∈ W m,∞ (M ) we can find a function ϕ ∈ W m,∞ (M ) satisfying (T − λI)ϕ = f

and

on M ,

kϕkm,∞ ≤ Cm kf km,∞ .

Here Cm > 0 is a constant independent of f .

(4) Therefore, Theorem 2.1 follows from Claims I and II by a well-known interpolation argument, since the space C k+θ (M ) is a real interpolation space between the spaces W k,∞ (M ) and W k+1,∞ (M ): C k+θ (M ) = W k,∞ (M ), W k+1,∞(M ) θ,∞ . 2.4 Positive Borel Kernels and the Positive Maximum Principle. Let Ω be an open subset of Rn , and let Bloc (Ω) = the space of Borel-measurable functions in Ω which are bounded on compact subsets of Ω. Let B be the σ-algebra of all Borel sets in Ω. A positive Borel kernel on Ω is a mapping x 7−→ s(x, dy)

of Ω into the space of non-negative measures on B such that, for each X ∈ B, the function s(·, X) is Borel-measurable on Ω. Now we assume that a positive Borel kernel s(x, dy) satisfies the following two conditions: (NS1) s(x, {x}) = 0 for all x ∈ Ω. (NS2) For all non-negative functions f in C0 (Ω), the function Z x 7−→ s(x, dy)|y − x|2 f (y), x ∈ Ω, Ω

belongs to the space Bloc (Ω). Let σ(x, y) be a smooth function on Ω × Ω such that (a) 0 ≤ σ(x, y) ≤ 1 on Ω × Ω. (b) σ(x, y) = 1 in a neighborhood of the diagonal {(x, x) : x ∈ Ω} in Ω × Ω. (c) For any compact subset K of Ω, there exists a compact subset K ′ of Ω such that the functions σx (·) = σ(x, ·), x ∈ K, have their support in K ′ . Then, by using Taylor’s formula and condition (NS2) we can define a linear operator S : C02 (Ω) −→ Bloc (Ω) by the formula (2.6)

n X ∂u (x)(yi − xi ) Su(x) = s(x, dy) u(y) − σ(x, y) u(x) + ∂x i Ω i=1

Z

"

u ∈ C02 (Ω).

!#

,

The next three theorems give useful characterizations of linear operators A from C02 (Ω) into Bloc (Ω) which satisfy the positive maximum principle (cf. [BCP, Th´eor`eme I, Th´eor`eme II and Th´eor`eme III]):

38

KAZUAKI TAIRA

Theorem 2.6. Let A be a linear operator from C02 (Ω) into Bloc (Ω). Then the following two assertions are equivalent: (i) A : C02 (Ω) → Bloc (Ω) is continuous and satisfies the condition (2.7)

x0 ∈ Ω, u ∈ C02 (Ω), u ≥ 0 in Ω and x0 6∈ supp u =⇒ Au(x0 ) ≥ 0.

(ii) There exist a second-order differential operator P : C 2 (Ω) → Bloc (Ω) and a positive Borel kernel s(x, dy), having properties (NS1) and (NS2), such that the operator A is written in the form (2.8)

x ∈ Ω, u ∈ C02 (Ω).

Au(x) = P u(x) + Su(x),

Theorem 2.7. Let V be a linear subspace of C02 (Ω) which contains C0∞ (Ω). Assume that A is a linear operator from V into Bloc (Ω) and satisfies the condition (2.9)

x0 ∈ Ω, u ∈ V, u ≥ 0 in Ω and u(x0 ) = 0 =⇒ Au(x0 ) ≥ 0.

Then the operator A can be extended uniquely to a continuous linear operator A : C02 (Ω) −→ Bloc (Ω) which still satisfies condition (2.9) for all u ∈ C02 (Ω): (2.9′ )

x0 ∈ Ω, u ∈ C02 (Ω), u ≥ 0 in Ω and u(x0 ) = 0 =⇒ Au(x0 ) ≥ 0.

Theorem 2.8. Let A be a linear operator from C02 (Ω) into Bloc (Ω) of the form (2.8), where P : C 2 (Ω) → Bloc (Ω) is a second-order differential operator on Ω and the operator S is defined by formula (2.6), with a positive Borel kernel s(x, dy) having properties (NS1) and (NS2). Then we have the following assertions: (i) The operator A satisfies condition (2.7) if and only if the principal symbol of P is non-positive. (ii) The operator A satisfies the condition (PM)

x0 ∈ Ω, v ∈ C02 (Ω) and v(x0 ) = sup v ≥ 0 =⇒ Av(x0 ) ≤ 0 Ω

if and only if the principal symbol of P is non-positive and P satisfies the following two conditions: P 1(x) ≤ 0,

x ∈ Ω, Z A1 = P 1(x) + s(x, dy) [1 − σ(x, y)] ≤ 0, Ω

x ∈ Ω.

3. Proof of Theorem 1 In this section we prove Theorem 1. First, we reduce the problem of construction of Feller semigroups to the problem of unique solvability for the boundary value problem (α − W )u = f in D, (λ − L)u = ϕ

on ∂D,

and then prove existence theorems for Feller semigroups. Here α is a positive number and λ is a non-negative number.

FELLER SEMIGROUPS AND MARKOV PROCESSES

39

3.1 General Existence Theorem for Feller Semigroups. The purpose of this subsection is to give a general existence theorem for Feller semigroups (Theorem 3.14) in terms of boundary value problems, following Taira [Ta1, Section 9.6] (cf. [BCP], [SU]). First, we consider the following Dirichlet problem: For given functions f and ϕ defined in D and on ∂D, respectively, find a function u in D such that (D)

(α − W )u = f u=ϕ

in D, on ∂D,

where α > 0. The next theorem summarizes the basic facts about the Dirichlet problem in the framework of H¨ older spaces (see [BCP, Th´eor`eme XV]): Theorem 3.1. Let k be an arbitrary non-negative integer and 0 < θ < 1. For any f ∈ C k+θ (D) and any ϕ ∈ C k+2+θ (∂D), problem (D) has a unique solution u in C k+2+θ (D). Theorem 3.1 with k := 0 tells us that problem (D) has a unique solution u in (D) for any f ∈ C θ (D) and any ϕ ∈ C 2+θ (∂D), 0 < θ < 1. Therefore, we can C introduce linear operators 2+θ

G0α : C θ (D) −→ C 2+θ (D), and Hα : C 2+θ (∂D) −→ C 2+θ (D) as follows. (a) For any f ∈ C θ (D), the function G0α f ∈ C 2+θ (D) is the unique solution of the problem (3.1)

(α − W )G0α f = f G0α f = 0

in D, on ∂D.

(b) For any ϕ ∈ C 2+θ (∂D), the function Hα ϕ ∈ C 2+θ (D) is the unique solution of the problem (3.2)

(α − W )Hα ϕ = 0 in D, Hα ϕ = ϕ on ∂D.

The operator G0α is called the Green operator and the operator Hα is called the harmonic operator , respectively. Then we have the following results (cf. [Ta1, Lemmas 9.6.2 and 9.6.3]): Lemma 3.2. The operator G0α , α > 0, considered from C(D) into itself, is nonnegative and continuous with norm

0 0

G = G 1 = max G0 1(x) . α α α ∞ x∈D

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KAZUAKI TAIRA

Proof. Let f be an arbitrary function in C θ (D) such that f ≥ 0 on D. Then, by applying Theorem A.1 (the weak maximum principle) with W := W − α to the function −G0α f we obtain from formula (3.1) that G0α f ≥ 0 on D. This proves the non-negativity of G0α . Since G0α is non-negative, we have, for all f ∈ C θ (D), −G0α kf k∞ ≤ G0α f ≤ G0α kf k∞

on D.

This implies the continuity of G0α with norm kG0α k = kG0α 1k∞ . The proof is complete.

Lemma 3.3. The operator Hα , α > 0, considered from C(∂D) into C(D), is non-negative and continuous with norm kHα k = kHα 1k∞ = max Hα 1(x) . x∈D

More precisely, we have the following (see [BCP, Proposition III.1.6]): Theorem 3.4. (i) (a) The operator G0α , α > 0, can be uniquely extended to a non-negative, bounded linear operator on C(D) into itself, denoted again by G0α , with norm (3.3)

0 0

Gα = Gα 1 ≤ 1 . ∞ α

(b) For any f ∈ C(D), we have

G0α f ∂D = 0.

(c) For all α, β > 0, the resolvent equation holds: (3.4)

G0α f − G0β f + (α − β)G0α G0β f = 0,

f ∈ C(D).

(d) For any f ∈ C(D), we have (3.5)

lim αG0α f (x) = f (x),

α→+∞

x ∈ D.

Furthermore, if f |∂D = 0, then this convergence is uniform in x ∈ D, that is, (3.5′ )

lim αG0α f = f

α→+∞

in C(D).

(e) The operator G0α maps C k+θ (D) into C k+2+θ (D) for any non-negative integer k.

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(ii) (a′ ) The operator Hα , α > 0, can be uniquely extended to a non-negative, bounded linear operator on C(∂D) into C(D), denoted again by Hα , with norm kHα k ≤ 1. (b′ ) For any ϕ ∈ C(∂D), we have Hα ϕ|∂D = ϕ. (c′ ) For all α, β > 0, we have (3.6)

Hα ϕ − Hβ ϕ + (α − β)G0α Hβ ϕ = 0,

ϕ ∈ C(∂D).

(d′ ) For any ϕ ∈ C(∂D), we have lim Hα ϕ(x) = 0,

α→+∞

x ∈ D.

(e′ ) The operator Hα maps C k+2+θ (∂D) into C k+2+θ (D) for any non-negative integer k. Proof. (i) (a) Making use of Friedrichs’ mollifiers, we find that the space C θ (D) is dense in C(D) and further that non-negative functions can be approximated by non-negative smooth functions. Hence, by Lemma 3.2 it follows that the operator G0α : C θ (D) → C 2+θ (D) can be uniquely extended to a non-negative, bounded linear operator G0α : C(D) → C(D) with norm kG0α k = kG0α 1k∞ . Further, since the function G0α 1 satisfies the conditions (W − α)G0α 1 = −1 in D, G0α 1 = 0 on ∂D, by applying Theorem A.2 with W := W − α we obtain that kG0α k = kG0α 1k∞ ≤

1 . α

(b) This follows from formula (3.1), since the space C θ (D) is dense in C(D) and the operator G0α : C(D) → C(D) is bounded. (c) We find from the uniqueness theorem for problem (D) (Theorem 3.2) that equation (3.4) holds for all f ∈ C θ (D). Hence it holds for all f ∈ C(D), since the space C θ (D) is dense in C(D) and the operators G0α are bounded. (d) First, let f be an arbitrary function in C θ (D) satisfying f |∂D = 0. Then it follows from the uniqueness theorem for problem (D) (Theorem 3.2) that we have, for all α, β, f − αG0α f = G0α ((β − W )f ) − βG0α f. Thus we have, by estimate (3.3), kf − αG0α f k∞ ≤

β 1 k(β − W )f k∞ + kf k∞ , α α

so that lim kf − αG0α f k∞ = 0.

α→+∞

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KAZUAKI TAIRA

Now let f be an arbitrary function in C(D) satisfying f |∂D = 0. By means of mollifiers, we can find a sequence {fj } in C θ (D) such that fj −→ f in C(D) as j → ∞, fj = 0

on ∂D.

Then we have, by estimate (3.3), kf − αG0α f k∞ ≤ kf − fj k∞ + kfj − αG0α fj k∞ + kαG0α fj − αG0α f k∞ ≤ 2kf − fj k∞ + kfj − αG0α fj k∞ ,

and hence lim sup kf − αG0α f k∞ ≤ 2kf − fj k∞ . α→+∞

This proves assertion (3.5′ ), since kf − fj k∞ → 0 as j → ∞. To prove assertion (3.5), let f be an arbitrary function in C(D) and x an arbitrary point of D. Take a function ψ ∈ C(D) such that 0 ≤ ψ ≤ 1 on D, ψ=0 in a neighborhood of x, ψ=1 near the set ∂D. Then it follows from the non-negativity of G0α and estimate (3.3) that

(3.7)

0 ≤ αG0α ψ(x) + αG0α (1 − ψ)(x) = αG0α 1(x) ≤ 1.

However, by applying assertion (3.5′ ) to the function 1 − ψ we have lim αG0α (1 − ψ)(x) = (1 − ψ)(x) = 1.

α→+∞

In view of inequalities (3.7), this implies that lim αG0α ψ(x) = 0.

α→+∞

Thus, since −kf k∞ ψ ≤ f ψ ≤ kf k∞ ψ on D, it follows that |αG0α (f ψ)(x)| ≤ kf k∞ αG0α ψ(x) → 0

as α → +∞.

Therefore, by applying assertion (3.5′ ) to the function (1 − ψ)f we obtain that f (x) = ((1 − ψ)f ) (x) = lim αG0α ((1 − ψ)f ) (x) = lim αG0α f (x). α→+∞

α→+∞

(ii) (a′ ) Since the space C 2+θ (∂D) is dense in C (∂D), by Lemma 3.3 it follows that the operator Hα : C 2+θ (∂D) → C 2+θ (D) can be uniquely extended to a nonnegative, bounded linear operator Hα : C (∂D) → C(D). Further, by applying Theorem A.2 with W := W − α we have kHα k = kHα 1k∞ = 1.

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(b′ ) This follows from formula (3.2), since the space C 2+θ (∂D) is dense in C (∂D) and the operator Hα : C (∂D) → C(D) is bounded. (c′ ) We find from the uniqueness theorem for problem (D) that formula (3.6) holds for all ϕ ∈ C 2+θ (∂D). Hence it holds for all ϕ ∈ C (∂D), since the space C 2+θ (∂D) is dense in C (∂D) and the operators G0α and Hα are bounded. The proof of Theorem 3.4 is now complete. Now we consider the following boundary value problem (∗) in the framework of the spaces of continuous functions: (α − W )u = f in D, (∗) Lu = 0 on ∂D. To do this, we introduce three operators associated with problem (∗). (I) First, we introduce a linear operator W : C(D) −→ C(D) as follows. (a) The domain D(W ) of W is the space C 2+θ (D). (b) W u = P u + Sr u for all u ∈ D(W ). Then we have the following (cf. [Ta1, Lemma 9.6.5]): Lemma 3.5. The operator W has its minimal closed extension W in the space C(D). Proof. We apply part (i) of Theorem 1.5 to the operator W . Assume taht a function u ∈ C 2 (D) takes a positive maximum at a point x0 of D: u(x0 ) = max u(x) > 0. x∈D

Then it follows that ∂u (x0 ) = 0, 1 ≤ i ≤ N, ∂xi N X ∂ 2u aij (x0 ) (x0 ) ≤ 0, ∂x ∂x i j i,j=1 since the matrix (aij (x)) is positive definite. Hence we have P u(x0 ) =

N X

aij (x0 )

i,j=1

Sr u(x0 ) =

Z

D

∂ 2u (x0 ) + c(x0 )u(x0 ) ≤ 0, ∂xi ∂xj

s(x0 , y) (u(y) − u(x0 )) dy ≤ 0.

This implies that the operator W = P + Sr satisfies condition (β) of Theorem 1.5 with K0 := D and K := D. Therefore, Lemma 3.5 follows from an application of the same theorem.

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Remark 3.6. Since the injection: C(D) → D ′ (D) is continuous, we have the formula N X

N X ∂ 2u ∂u W u(x) = a (x) (x) + bi (x) (x) + c(x)u(x) ∂x ∂x ∂x i j i i,j=1 i=1 Z N X ∂u + s(x, y) u(y) − u(x) − (x) dy, (yj − xj ) ∂x j D j=1 ij

where the right-hand side is taken in the sense of distributions. The extended operators G0α : C(D) −→ C(D) and Hα : C(∂D) −→ C(D), α > 0, still satisfy formulas (3.1) and (3.2) respectively in the following sense (cf. [Ta1, Lemma 9.6.7 and Corollary 9.6.8]): Lemma 3.7. (i) For any f ∈ C(D), we have

G0α f ∈ D(W ),

(αI − W )G0α f = f

in D.

(ii) For any ϕ ∈ C(∂D), we have

Hα ϕ ∈ D(W ),

(αI − W )Hα ϕ = 0

in D.

Here D(W ) is the domain of the closed extension W . Proof. (i) Choose a sequence {fj } in C θ (D) such that fj → f in C(D) as j → ∞. Then it follows from the boundedness of G0α that G0α fj −→ G0α f

in C(D),

and also (α − W )G0α fj = fj −→ f Hence we have

G0α f ∈ D(W ),

(αI − W )G0α f = f

in C(D).

in D.

since the operator W : C(D) → C(D) is closed. (ii) Similarly, part (ii) is proved, since the space C 2+θ (∂D) is dense in C(∂D) and the operator Hα : C(∂D) → C(D) is bounded. Corollary 3.8. Every u in D(W ) can be written in the following form: (3.8)

u = G0α (αI − W )u + Hα (u|∂D ),

α > 0.

Proof. We let w = u − G0α (αI − W )u − Hα (u|∂D ).

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Then it follows from Lemma 3.7 that the function w is in D(W ) and satisfies (αI − W )w = 0 in D, w=0

on ∂D.

Thus, in view of Remark 3.6 we can apply Theorem 3.1 to obtain that w = 0. This proves formula (3.8). (II) Secondly, we introduce a linear operator LG0α : C(D) −→ C(∂D) as follows. 0 (a) The domain D LG of LG0α is the space C θ (D). α (b) LG0α f = L G0α f for all f ∈ D LG0α . Then we have the following (cf. [Ta1, Lemma 9.6.9]): Lemma 3.9. The operator LG0α , α > 0, can be uniquely extended to a nonnegative, bounded linear operator LG0α : C(D) −→ C(∂D).

Proof. Let f be an arbitrary function in D(LG0α ) such that f ≥ 0 on D. Then we have 0 2 Gα f ∈ C (D), G0α f ≥ 0 on D, 0 Gα f = 0 on ∂D, and hence

∂ (G0 f ) − δ(x′ )AG0α f ∂n α ∂ = µ(x′ ) (G0α f ) + δ(x′ )f ≥ 0 on ∂D. ∂n

LG0α f = µ(x′ )

This proves that the operator LG0α is non-negative. By the non-negativity of LG0α , we have, for all f ∈ D(LG0α ), −LG0α kf k∞ ≤ LG0α f ≤ LG0α kf k∞

on ∂D.

This implies the boundedness of LG0α with norm kLG0α k = kLG0α 1k∞ . Recall that the space C θ (D) is dense in C(D) and that non-negative functions can be approximated by non-negative C ∞ functions. Hence we find that the operator LG0α can be uniquely extended to a non-negative, bounded linear operator LG0α : C(D) → C(∂D). The next lemma states a fundamental relationship between the operators LG0α and LG0β for α, β > 0 (cf. [Ta1, Lemma 9.6.10]):

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KAZUAKI TAIRA

Lemma 3.10. For any f ∈ C(D), we have LG0α f − LG0β f + (α − β)LG0α G0β f = 0,

(3.9)

α, β > 0.

Proof. Choose a sequence {fj } in C θ (D) such that fj → f in C(D) as j → ∞. Then, by using the resolvent equation (3.4) with f := fj we have LG0α fj − LG0β fj + (α − β)LG0α G0β fj = 0. Hence formula (3.9) follows by letting j → ∞, since the operators LG0α , LG0β and G0β are all bounded. (III) Finally, we introduce a linear operator LHα : C(∂D) −→ C(∂D) as follows. (a) The domain D (LHα ) of LHα is the space C 2+θ (∂D). (b) LHα ψ = L (Hα ψ) for all ψ ∈ D (LHα ). Then we have the following (cf. [Ta1, Lemma 9.6.11]):

Lemma 3.11. The operator LHα , α > 0, has its minimal closed extension LHα in the space C(∂D). Proof. We apply part (i) of Theorem 1.5 to the operator LHα . To do this, it suffices to show that the operator LHα satisfies condition (β ′ ) with K := ∂D (or condition (β) with K := K0 = ∂D) of the same theorem. Assume that a function ψ in D(LHα ) = C 2+θ (∂D) takes its positive maximum at some point x′ of ∂D. Since the function Hα ψ is in C 2+θ (D) and satisfies

(W − α)Hα ψ = 0 Hα ψ = ψ

in D, on ∂D,

by applying Theorem A.1 (the weak maximum principle) with W := W − α to the function Hα ψ we find that the function Hα ψ takes its positive maximum at x′ ∈ ∂D. Thus we can apply Lemma A.2 with Σ3 := ∂D to obtain that ∂ (Hα ψ)(x′ ) < 0. ∂n

(3.10) Hence we have LHα ψ(x′ ) =

N−1 X

i,j=1

αij (x′ )

∂ ∂ 2ψ (x′ ) + µ(x′ ) (Hα ψ)(x′ ) ∂xi ∂xj ∂n

+ γ(x′ )ψ(x′ ) − αδ(x′ )ψ(x′ )

≤ 0.

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47

This verifies condition (β ′ ) of Theorem 1.5. Therefore, Lemma 3.11 follows from an application of the same theorem. Remark 3.12. The operator LHα enjoys the following property: (3.11) If a function ψ in the domain D LHα takes its positive maximum at some point x′ of ∂D, then we have LHα ψ(x′ ) ≤ 0. The next lemma states a fundamental relationship between the operators LHα and LHβ for α, β > 0 (cf. [Ta1, Lemma 9.6.13]): Lemma 3.13. The domain D LHα of LHα does not depend on α > 0; so we denote by D the common domain. Then we have (3.12)

LHα ψ − LHβ ψ + (α − β)LG0α Hβ ψ = 0,

α, β > 0, ψ ∈ D.

Proof. Let ψ be an arbitrary function in D(LHβ ), and choose a sequence {ψj } in D(LHβ ) = C 2+θ (∂D) such that ψj −→ ψ in C(∂D), LHβ ψj −→ LHβ ψ

in C(∂D).

Then it follows from the boundedness of Hβ and LG0α that LG0α (Hβ ψj ) = LG0α (Hβ ψj ) −→ LG0α (Hβ ψ) in C(∂D). Therefore, by using formula (3.6) with ϕ := ψj we obtain that LHα ψj = LHβ ψj − (α − β)LG0α (Hβ ψj )

−→ LHβ ψ − (α − β)LG0α (Hβ ψ) in C(∂D).

This implies that

ψ ∈ D(LHα ),

LHα ψ = LHβ ψ − (α − β)LG0α (Hβ ψ).

Conversely, by interchanging α and β we have D(LHα ) ⊂ D(LHβ ), and so D(LHα ) = D(LHβ ). This proves the lemma. Now we can give a general existence theorem for Feller semigroups on ∂D in terms of boundary value problem (∗). The next theorem tells us that the operator LHα is the infinitesimal generator of some Feller semigroup on ∂D if and only if problem (∗) is solvable for sufficiently many functions ϕ in the space C(∂D) (cf. [Ta1, Theorem 9.6.15]):

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KAZUAKI TAIRA

Theorem 3.14. (i) If the operator LHα , α > 0, is the infinitesimal generator of a Feller semigroup on ∂D, then, for each constant λ > 0, the boundary value problem

(∗′ )

(α − W )u = 0 (λ − L)u = ϕ

in D, on ∂D

has a solution u ∈ C 2+θ (D) for any ϕ in some dense subset of C(∂D). (ii) Conversely, if, for some constant λ ≥ 0, problem (∗′ ) has a solution u ∈ 2+θ C (D) for any ϕ in some dense subset of C(∂D), then the operator LHα is the infinitesimal generator of some Feller semigroup on ∂D. Proof. (i) If the operator LHα generates a Feller semigroup on ∂D, by applying part (i) of Theorem 1.5 with K := ∂D to the operator LHα we obtain that R λI − LHα = C(∂D)

for each λ > 0.

This implies that the range R (λI − LHα ) is a dense subset of C(∂D) for each λ > 0. However, if ϕ ∈ C(∂D) is in the range R (λI − LHα ), and if ϕ = (λI − LHα ) ψ with ψ ∈ C 2+θ (∂D), then the function u = Hα ψ ∈ C 2+θ (D) is a solution of problem (∗)0 . This proves part (i). (ii) We apply part (ii) of Theorem 1.5 with K := ∂D to the operator LHα . To do this, it suffices to show that the operator LHα satisfies condition (γ) of the same theorem, since it satisfies condition (β ′ ), as is shown in the proof of Lemma 3.11. By the uniqueness theorem for problem (D′ ), it follows that any function u ∈ C 2+θ (D) which satisfies the equation (α − W )u = 0

in D

can be written in the form: u = Hα (u|∂D ) ,

u|∂D ∈ C 2+θ (∂D) = D (LHα ) .

Thus we find that if there exists a solution u ∈ C 2+θ (D) of problem (∗)0 for a function ϕ ∈ C(∂D), then we have (λI − LHα ) (u|∂D ) = ϕ, and so ϕ ∈ R (λI − LHα ) . Therefore, if, for some constant λ ≥ 0, problem (∗)0 has a solution u ∈ C 2+θ (D) for any ϕ in some dense subset of C(∂D), then the range R (λI − LHα ) is dense in C(∂D). This verifies condition (γ) (with α0 := λ) of Theorem 1.5. Hence part (ii) follows from an application of the same theorem. Theorem 3.14 is proved. We conclude this subsection by giving a precise meaning to the boundary conditions Lu for functions u in the domain D(W ).

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We let D(L) = u ∈ D(W ) : u|∂D ∈ D ,

where D is the common domain of the operators LHα , α > 0. It should be noticed that the domain D(L) contains the space C 2+θ (D), since C 2+θ (∂D) = D (LHα ) ⊂ D. Corollary 3.8 tells us that every function u in D(L) ⊂ D(W ) can be written in the form (3.8) u = G0α (αI − W )u + Hα (u|∂D ) , α > 0. Then we define

Lu = LG0α (αI − W )u + LHα (u|∂D ) .

(3.13)

The next lemma justifies definition (3.13) of Lu for all u ∈ D(L) (cf. [Ta1, Lemma 9.6.16]): Lemma 3.15. The right-hand side of formula (3.13) depends only on u, not on the choice of expression (3.8). Proof. Assume that u = G0α = G0β

αI − W u + Hα (u|∂D ) βI − W u + Hβ (u|∂D ) ,

where α > 0, β > 0. Then it follows from formula (3.9) with f := αI − W u and formula (3.12) with ψ := u|∂D that LG0α αI − W u + LHα (u|∂D ) (3.14) = LG0β αI − W u − (α − β)LG0α G0β αI − W u + LHβ (u|∂D ) − (α − β)LG0α Hβ (u|∂D ) = LG0β (βI − W )u + LHβ (u|∂D ) n o + (α − β) LG0β u − LG0α G0β αI − W u − LG0α Hβ (u|∂D ) .

However, we obtain from formula (3.9) with f := u that (3.15) LG0β u − LG0α G0β αI − W u − LG0α Hβ (u|∂D ) = LG0β u − LG0α G0β βI − W u + Hβ (u|∂D ) + (α − β)G0β u = LG0β u − LG0α u − (α − β)LG0α G0β u

= 0.

Therefore, by combining formulas (3.14) and (3.15) we have LG0α αI − W u + LHα (u|∂D ) = LG0β βI − W u + LHβ (u|∂D ) . This proves Lemma 3.15.

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3.2 End of Proof of Theorem 1. The next theorem proves Theorem 1: Theorem 3.16. We define a linear operator A : C(D) −→ C(D) as follows (see formula (0.4)). (a) The domain D(A) of A is the set D(A) = u ∈ D(W ) : u|∂D ∈ D, Lu = 0 ,

(3.16)

where D is the common domain of the operators LHα , α > 0. (b) Au = W u for all u ∈ D(A). If the boundary condition L is transversal on the boundary ∂D, then the operator A is the infinitesimal generator of some Feller semigroup on D, and the Green −1 operator Gα = (αI − A) , α > 0, is given by the formula (3.17)

−1 Gα f = G0α f − Hα LHα LG0α f ,

f ∈ C(D).

Remark 3.17. Intuitively, formula (3.17) asserts that if the boundary condition L is transversal on the boundary ∂D, then we can “piece together” a Markov process (Feller semigroup) on the boundary ∂D with W -diffusion in the interior D to construct a Markov process (Feller semigroup) on the closure D = D ∪ ∂D (see Figure 0.8). Proof of Theorem 3.16. We apply part (ii) of Theorem 1.3 to the operator A defined by formula (3.16). The proof is divided into several steps. (1) We let L0 u(x′ ) N−1 X ∂u ′ ∂ 2u ′ (x ) + β i (x′ ) (x ) = α (x ) ∂x ∂x ∂x i j i i=1 i,j=1 N−1 X

ij

′

∂u + γ(x′ )u(x′ ) + µ(x′ ) (x′ ) ∂n Z N−1 X ∂u ′ ′ (x ) dy + r(x′ , y ′ ) u(y ′ ) − u(x′ ) − (yj − xj ) ∂xj ∂D j=1 Z N−1 X ∂u ′ + t(x′ , y) u(y) − u(x′ ) − (yj − xj ) (x ) dy, ∂xj D j=1

and consider the term −δ(x′ )(W u|∂D ) in Lu as a term of “perturbation” of L0 u: Lu = L0 u − δ(x′ ) (W u|∂D ) .

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It is easy to see that the operator LHα can be decomposed as follows: LHα = L0 Hα − αδ(x′ )I. First, we prove that For all α > 0, the operator L0 Hα generates a Feller semigroup on the boundary ∂D. To do this, we remark that L0 Hα ϕ(x′ ) =

N−1 X

αij (x′ )

i,j=1

N−1 X ∂ 2ϕ ∂ϕ ′ β i (x′ ) (x′ ) + (x ) ∂xi ∂xj ∂xi i=1

∂ + γ(x′ )ϕ(x′ ) + µ(x′ ) (Hα ϕ)(x′ ) ∂n Z N−1 X ∂ϕ ′ ′ (yj − xj ) + r(x′ , y ′ ) ϕ(y ′ ) − ϕ(x′ ) − (x ) dy ∂xj ∂D j=1 Z N−1 X ∂ϕ ′ (x ) dy. + t(x′ , y) Hα ϕ(y) − ϕ(x′ ) − (yj − xj ) ∂x j D j=1 However, we have the following results: (a) The operator N−1 X ∂ϕ ′ ∂ 2ϕ ′ β i (x′ ) (x ) + (x ) + γ(x′ )ϕ(x′ ) α (x ) ϕ(x ) 7−→ ∂x ∂x ∂x i j i i=1 i,j=1 N−1 X

′

ij

′

is a second-order degenerate elliptic differential operator on ∂D with non-positive principal symbol, and γ(x′ ) ≤ 0 on ∂D. (b) The operator ∂ ϕ(x′ ) 7−→ µ(x′ ) (Hα ϕ)(x′ ) ∂n is a classical, pseudo-differential operator of order 1 on ∂D (see [Ho], [RS]). Moreover, it should be noticed that (3.18)

x′0 ∈ ∂D, ϕ ∈ C 2 (∂D), ϕ ≥ 0 on ∂D and x′0 6∈ supp ϕ ∂ =⇒ (Hα ϕ)(x′0 ) ≥ 0. ∂n

Indeed, if we let u = Hα ϕ,

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KAZUAKI TAIRA

then we have

(α − W )u = 0 in D,

u=ϕ

on ∂D.

Since ϕ ≥ 0 on ∂D, it follows from an application of the maximum principle that Hα ϕ = u ≥ 0 in D. Hence this implies that

∂ (Hα ϕ)(x′0 ) ≥ 0, ∂n

since ϕ(x′0 ) = 0. (c) The operator ϕ(x′ ) 7−→

Z

∂D

r(x′ , y ′ ) ϕ(y ′ ) − ϕ(x′ ) −

N−1 X j=1

(yj − xj )

∂ϕ ′ ′ (x ) dy ∂xj

is a classical, pseudo-differential operator of order 2 − κ1 on ∂D. (d) The operator Z N−1 X ∂ϕ ′ (x ) dy ϕ(x′ ) 7−→ t(x′ , y) Hα ϕ(y) − ϕ(x′ ) − (yj − xj ) ∂x j D j=1

is a classical, pseudo-differential operator of order 2 −κ2 on the boundary ∂D, since 2−κ2 T ∈ L1,0 (RN ) has the transmission property with respect to the boundary ∂D (see [Bo], [RS]). (e) It should be noticed that x′0 ∈ ∂D, ϕ ∈ C 2 (∂D), ϕ ≥ 0 on ∂D and x′0 6∈ supp ϕ =⇒ L0 Hα ϕ(x′0 ) ≥ 0.

Indeed, we have, by assertion (3.18) L0 Hα ϕ(x′0 )

=

µ(x′0 ) +

Z

D

≥ 0.

∂ (Hα ϕ)(x′0 ) + ∂n

Z

∂D

r(x′0 , y ′ )ϕ(y ′ ) dy ′

t(x′0 , y)Hα ϕ(y) dy

By virtue of Theorem 2.6, this proves that the operator L0 Hα may be written in the form (2.8). (f) Finally, since the function Hα 1 takes its positive maximum 1 only on the boundary ∂D, it follows from an application of the boundary point lemma (see Appendix, Lemma A.2) that (3.19)

∂ (Hα 1) < 0 on ∂D. ∂n

FELLER SEMIGROUPS AND MARKOV PROCESSES

53

Hence we have, by transversality condition (0.3), L0 Hα 1(x′ ) ∂ = γ(x ) + µ(x ) (Hα 1)(x′ ) + ∂n ≤ 0 on ∂D. ′

′

Z

D

t(x′ , y) [Hα 1(y) − 1] dy

Thus, by applying Theorem 2.1 to the operator L0 Hα we find that (3.20)

If λ > 0 is sufficiently large, then the range R(L0 Hα − λI)

contains the space C 2+θ (∂D).

This implies that the range R(L0 Hα −λI) is a dense subset of C(∂D). Therefore, by applying part (ii) of Theorem 3.14 to the operator L0 we obtain that the operator L0 Hα is the infinitesimal generator of some Feller semigroup on ∂D, for any α > 0. (2) Next we prove that For all α > 0, the operator LHα = L0 Hα − αδ(x′ )I generates a Feller semigroup on ∂D. It should be noticed that the operator −αδ(x′ )I is a bounded linear operator on the space C(∂D) into itself, and satisfies condition (β ′ ) of Theorem 1.5, since α > 0 and δ(x′ ) ≥ 0 on ∂D. Therefore, by applying Corollary 1.6 with A = L0 H α ,

C = −αδ(x′ )I,

we obtain that the operator LHα = L0 Hα − αδ(x′ )I is the infinitesimal generator of a Feller semigroup on ∂D, for any α > 0. (3) Now we prove that (3.21)

The equation LHα ψ = ϕ has a unique solution ψ in D(LHα ) for any ϕ ∈ C(∂D); hence the inverse LHα

−1

of LHα can be defined on the whole space C(∂D).

Further the operator −LHα

−1

is non-negative and bounded on C(∂D).

We have, by inequality (3.19) and transversality condition (0.3), LHα 1(x′ ) ∂ = γ(x ) + µ(x ) (Hα 1)(x′ ) − αδ(x′ ) + ∂n < 0 on ∂D, ′

′

Z

D

t(x′ , y) [Hα 1(y) − 1] dy

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KAZUAKI TAIRA

and so ℓα = − sup LHα 1(x′ ) > 0. x′ ∈∂D

Further, by using Corollary 1.4 with K := ∂D, A := LHα and c := ℓα we obtain that the operator LHα + ℓα I is the infinitesimal generator of some Feller semigroup on ∂D. Therefore, since ℓα > 0, it follows from an application of part (i) of Theorem 1.3 with A := LHα + ℓα I that the equation −LHα ψ = ℓα I − (LHα + ℓα I) ψ = ϕ

has a unique solution ψ ∈ D(LHα ) for any ϕ ∈ C(∂D), and further that the −1 −1 is non-negative and bounded on the operator −LHα = ℓα I − (LHα + ℓα I) space C(∂D) with norm

−1 −1

−LHα = ℓα I − (LHα + ℓα I)

∞

≤

1 . ℓα

(4) By assertion (3.21), we can define the right-hand side of formula (3.17) for all α > 0. We prove that Gα = (αI − A)

(3.22)

−1

,

α > 0.

In view of Lemmas 3.7 and 3.13, it follows that we have, for all f ∈ C(D),

and that

−1 0 0f G f = G f − H LH LG ∈ D(W ), α α α α α −1 LG0α f ∈ D LHα = D, Gα f |∂D = −LHα LG f = LG0 f − LH LH −1 LG0 f = 0, α α α α α (αI − W )Gα f = f.

This proves that

Gα f ∈ D(A),

(αI − A)Gα f = f,

that is, (αI − A)Gα = I

on C(D).

Therefore, in order to prove formula (3.22) it suffices to show the injectivity of the operator αI − A for α > 0. Assume that u ∈ D(A) and (αI − A)u = 0. Then, by Corollary 3.8, the function u can be written as u = Hα (u|∂D ) ,

u|∂D ∈ D = D LHα .

FELLER SEMIGROUPS AND MARKOV PROCESSES

55

Thus we have LHα (u|∂D ) = Lu = 0. In view of assertion (3.21), this implies that u|∂D = 0, so that u = Hα (u|∂D ) = 0

in D.

(5) The non-negativity of Gα , α > 0, follows immediately from formula (3.17), −1 since the operators G0α , Hα , −LHα and LG0α are all non-negative. (6) We prove that the operator Gα is bounded on the space C(D) with norm kGα k ≤

(3.23)

1 , α

α > 0.

To do this, it suffices to show that (3.23′ )

Gα 1 ≤

1 α

on D.

since Gα is non-negative on C(D). First, it follows from the uniqueness property of solutions of problem (D′ ) that αG0α 1 + Hα 1 = 1 + G0α (W 1) on D.

(3.24)

In fact, the both sides have the same boundary value 1 and satisfy the same equation: (α − W )u = α in D. By applying the operator L to the both hand sides of equality (3.24), we obtain that − LHα 1(x′ )

= −L1(x′ ) − LG0α (W 1)(x′ ) + αLG0α 1(x′ ) Z ′ ′ ∂ 0 ′ = −γ(x ) − µ(x ) (Gα (W 1))(x ) − t(x′ , y)G0α (W 1)(y)dy + αLG0α 1(x′ ) ∂n D ≥ αLG0α 1(x′ ) on ∂D,

since we have the assertions W 1(x) = P 1(x) + Sr 1(x) = c(x) +

Z

D

G0α (W 1) ≤ 0

on D,

G0α (W 1)|∂D = 0

s(x, y) [1 − σ(x, y)] dy ≤ 0 in D,

on ∂D.

Hence we have, by the non-negativity of −LHα (3.25)

−LHα

−1

1 LG0α 1 ≤ α

−1

, on ∂D.

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KAZUAKI TAIRA

By using formula (3.17) with f := 1, inequality (3.25) and equality (3.24), we obtain that 1 −1 0 0 LGα 1 ≤ G0α 1 + Hα 1 Gα 1 = Gα 1 + Hα −LHα α 1 0 1 = + Gα (W 1) α α 1 ≤ on D, α since the operators Hα and G0α are non-negative and since W 1 ≤ 0 in D. (7) Finally, we prove that (3.26)

The domain D(A) is everywhere dense in the space C(D).

(7-1) Before the proof, we need some lemmas on the behavior of G0α , Hα and −1 −LHα as α → +∞ (see [BCP, Proposition III.1.6]; [Ta1, Lemmas 9.6.19 and 9.6.20]): Lemma 3.18. For all f ∈ C(D), we have (3.27) lim αG0α f + Hα (f |∂D ) = f α→+∞

in C(D).

Proof. Choose a constant β > 0 and let g = f − Hβ (f |∂D ). Then, by using formula (3.6) with ϕ := f |∂D we obtain that (3.28) αG0α g − g = αG0α f + Hα (f |∂D ) − f − βG0α Hβ (f |∂D ). However, we have, by estimate (3.3),

lim G0α Hβ (f |∂D ) = 0

α→+∞

in C(D),

and by assertion (3.5′ ): lim αG0α g = g

α→+∞

in C(D),

since g|∂D = 0. Therefore, formula (3.27) follows by letting α → +∞ in formula (3.28). Lemma 3.19. The function

∂ (Hα 1) ∂n diverges to −∞ uniformly and monotonically as α → +∞. Proof. First, formula (3.6) with ϕ := 1 gives that Hα 1 = Hβ 1 − (α − β)G0α Hβ 1.

FELLER SEMIGROUPS AND MARKOV PROCESSES

57

Thus, in view of the non-negativity of G0α and Hα it follows that α ≥ β =⇒ Hα 1 ≤ Hβ 1 on D. Since Hα 1|∂D = Hβ 1|∂D = 1, this implies that the functions ∂ (Hα 1) ∂n are monotonically non-increasing in α. Further, by using formula (3.5) with f := Hβ 1 we find that the function

β Hα 1(x) = Hβ 1(x) − 1 − α

αG0α Hβ 1(x)

converges to zero monotonically as α → +∞, for each interior point x of D. Now, for any given constant K > 0 we can construct a function u ∈ C 2 (D) such that (3.29a) (3.29b)

u = 1 on ∂D, ∂u ≤ −K on ∂D. ∂n

Indeed, it follows from Theorem 3.1 that, for any integer m > 0, the function m

u = (Hα0 1) ,

α0 > 0,

belongs to C 2+θ (D) and satisfies condition (3.29a). Further we have ∂ ∂u =m (Hα0 1) ∂n ∂n ∂ ≤ m sup (Hα0 1) (x′ ). x′ ∈∂D ∂n m

In view of inequality (3.19), this implies that the function u = (Hα0 1) satisfies condition (3.29b) for m sufficiently large. Take a function u ∈ C 2 (D) which satisfies conditions (3.29a) and (3.29b), and choose a neighborhood U of ∂D, relative to D, with smooth boundary ∂U such that (see Figure 3.1) (3.30)

u≥

1 2

on U .

58

KAZUAKI TAIRA

...................................................................... ................ ............ . . . . . . . . . . .......... .... . . . . . . ........ . .... . . .. .... .... .... .... .... .... .... .... .... .... . . ....... . . . . . . . . . . . .... .... . . . . . . . . . . . .... .. ..... . .. ... . .... . ... ..... .. ... . ∂D ....... . . .. ... .. . .. . . .... . . . . .. . ... . . . ... ... .... .... U . . .. .. ... . . .... .... . ... . .. . ... ... . ... .. .. .... .... .... .. . . . . ... . .. .... ... ... .. ... . . . . . . . . . . . . . .. . . . U . . .. . . . . . . . . ... ... U . . .. .... ... ..... ... .. ... . .... . . . . . . . .. .. .... .. ... ... .... .. .. ... .. .. . ... . . . . ... . . . . . . . .... ... ... ... ... .. ... .... .... ... .. . . .... .... .... ... .... ...... ... ... . . . . ... . . . . ... .. .... .... . .... .... ... .. ∂D .... . .... .... . . . .... . . ∂D . .. .... .... .... .... .... ... .... ..... .... . . . . .... . . . . .. ... ... ...... ... .... .. . .... .. .... .... .... ....... ...... .... .... .... ... .. .... .. . . . .... .... .... .... .... .... .. . . ......... . ..... .......... ......... . . . ............. . . . . . . . . ..... ..................... ....................................................... Figure 3.1 Recall that the function Hα 1 converges to zero in D monotonically as α → +∞. Since u|∂D = Hα 1|∂D = 1, by using Dini’s theorem we can find a constant α > 0 (depending on u and hence on K) such that (3.31a) (3.31b)

Hα 1 ≤ u on ∂U \ ∂D,

α > 2kAuk∞ .

It follows from inequalities (3.30) and (3.31b) that (W − α)(Hα 1 − u) = αu − W u α ≥ − kW uk∞ 2 > 0 in U . Thus, by applying Theorem A.1 with W := W −α to the function Hα 1−u we obtain that the function Hα 1 − u may take its positive maximum only on the boundary ∂U . However, conditions (3.29a) and (3.31a) imply that Hα 1 − u ≤ 0 on ∂U = (∂U \ ∂D) ∪ ∂D. Therefore, we have Hα 1 ≤ u on U = U ∪ ∂U , and hence

∂ ∂u (Hα 1) ≤ ≤ −K ∂n ∂n

since u|∂D = Hα 1|∂D = 1. The proof of Lemma 3.19 is complete.

on ∂D,

FELLER SEMIGROUPS AND MARKOV PROCESSES

59

Corollary 3.20. If the boundary condition L is transversal on the boundary ∂D, then we have −1 lim k − LHα k = 0. α→+∞

Proof. We recall that LHα 1(x′ ) Z ∂ ′ ′ = γ(x ) + µ(x ) (Hα 1)(x ) − αδ(x ) + t(x′ , y) [Hα 1(y) − 1] dy ∂n D Z ′ ∂ ′ ′ ≤ µ(x ) (Hα 1)(x ) − αδ(x ) + t(x′ , y) [Hα 1(y) − 1] dy. ∂n D ′

′

However, it follows from an application of Beppo-Levi’s theorem that Z Z ′ lim t(x , y) [Hα 1(y) − 1] dy = − t(x′ , y) dy, α→+∞

D

D

since the function Hα 1 converges to zero in D monotonically as α → +∞. Hence we obtain from Lemma 3.19 that if the boundary condition L is transversal on the boundary ∂D, that is, if we have Z t(x′ , y)dy = +∞ if µ(x′ ) = δ(x′ ) = 0, D

then the function LHα 1 diverges to −∞ monotonically as α → +∞. By Dini’s theorem, this convergence is uniform in x′ ∈ ∂D. Hence the function 1 LHα 1(x′ ) converges to zero uniformly in x′ ∈ ∂D as α → +∞. This gives that

−1 −1

= −LH 1 −LH

α α ∞

1

≤

LHα 1 −→ 0 as α → +∞, ∞

since we have

−LHα 1(x′ ) 1

(−LHα 1(x′ )) , 1= ≤

′ |LHα 1(x )| LHα 1 ∞

x′ ∈ ∂D.

(7-2) Proof of assertion (3.26) In view of formula (3.22) and inequality (3.23), it suffices to prove that (3.32)

lim kαGα f − f k∞ = 0,

α→+∞

since the space C 2+θ (D) is dense in C(D).

f ∈ C 2+θ (D),

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KAZUAKI TAIRA

First, we remark that kαGα f − f k∞

−1

0

0 = αGα f − αHα LHα LGα f − f ∞

≤ αG0α f + Hα (f |∂D ) − f ∞

−1

+ −αHα LHα LG0α f − Hα (f |∂D ) ∞

0

≤ αGα f + Hα (f |∂D ) − f ∞

−1

+ −αLHα LG0α f − f |∂D . ∞

Thus, in view of formula (3.27) it suffices to show that i h −1 0 LGα f − f |∂D = 0 in C(∂D). (3.33) lim −αLHα α→+∞

Take a constant β such that 0 < β < α, and write f = G0β g + Hβ ϕ,

where (cf. formula (3.8)):

g = (β − W )f ∈ C θ (D),

ϕ = f |∂D ∈ C 2+θ (∂D).

Then, by using equations (3.4) (with f := g) and (3.6) we obtain that G0α f = G0α G0β g + G0α Hβ ϕ =

1 G0β g − G0α g + Hβ ϕ − Hα ϕ . α−β

Hence we have

−1

0 LGα f − f |∂D

−αLHα ∞

α

α −1 0 0 LGβ g − LGα g + LHβ ϕ + = ϕ − ϕ

α − β −LHα

α−β ∞

α −1 ≤

−LHα · LG0β g + LHβ ϕ ∞ α−β

β α −1

kϕk∞ . +

−LHα · LG0α ∞ · kgk∞ + α−β α−β

By Corollary 3.20, it follows that the first term on the last inequality converges to zero as α → +∞. For the second term, by using formula (3.4) with f := 1 and the non-negativity of G0β and LG0α we find that kLG0α k = kLG0α 1k∞

= kLG0β 1 − (α − β)LG0α G0β 1k∞

≤ kLG0β 1k∞ .

FELLER SEMIGROUPS AND MARKOV PROCESSES

61

Hence the second term also converges to zero as α → +∞. It is clear that the third term converges to zero as α → +∞. This completes the proof of assertion (3.33) and hence that of assertion (3.32). (8) Summing up, we have proved that the operator A, defined by formula (3.16), satisfies conditions (a) through (d) in Theorem 1.3. Hence it follows from an application of the same theorem that the operator A is the infinitesimal generator of some Feller semigroup on D. The proof of Theorem 3.16 and hence that of Theorem 1 is now complete. 4. Proof of Theorem 2 In this section we prove Theorem 2. First, we show that if condition (A) is satisfied, then the operator LHα is bijective in the framework of H¨older spaces. This is proved by applying Theorem 2.1 just as in the proof of Theorem 1. Therefore, we find that a unique solution u of problem (α − W )u = f in D, (∗∗) Lu = m(x′ ) Lν u + γ(x′ )u = 0 on ∂D can be expressed as follows: −1 ν ν u = Gα f = Gα f − Hα LHα (LGα f ) . This formula allows us to verify all the conditions of the generation theorems of Feller semigroups discussed in Subsection 1.2.

4.1 The Space C0 (D \ M ). First, we consider a one-point compactification K∂ = K ∪ {∂} of the space K = D \ M , where M = {x′ ∈ ∂D : m(x′ ) = 0}.

We say that two points x and y of D are equivalent modulo M if x = y or x, y ∈ M ; we then write x ∼ y. We denote by D/M the totality of equivalence classes modulo M . On the set D/M , we define the quotient topology induced by the projection q : D → D/M . It is easy to see that the topological space D/M is a one-point compactification of the space D \ M and that the point at infinity ∂ corresponds to the set M : K∂ = D/M, ............ ∂D .............. ........................ ...... ....... . . . . . .... ...... .... . . . . . . . ... ........... . . . . . . . ... . . . . . . . . . . . . . .. . . . . . .... ... .... ..... ... . D .. ... ... ... . . ... .. .... .... ... ..... . . . . ........ ..................................................................................

..............................

................................................

∂ = M.

........................................... ................. ...... ............ . .... . . . . . . . . ... ........ . . . ... . . . . . . ... . . . . . . . .. .... . . . . . . . . . .. D/M ... .. .... ... . ... .. .... ... ...... ... . ......... . .... ............ ...................................................................

M Figure 4.1

• ∂

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KAZUAKI TAIRA

Furthermore, we have the following isomorphism: C(K∂ ) ∼ = u ∈ C(D) : u is constant on M .

(4.1)

Now we introduce a closed subspace of C(K∂ ) as in Subsection 1.1: C0 (K) = {u ∈ C(K∂ ) : u(∂) = 0} . Then we have, by assertion (4.1), C0 (K) ∼ = C0 (D \ M ) = u ∈ C(D) : u = 0 on M .

(4.2)

4.2 End of Proof of Theorem 2. We shall apply part (ii) of Theorem 1.3 to the operator W defined by formula (0.5). First, we simplify the boundary condition Lu = 0 on ∂D. If conditions (A) and (H) are satisfied, then we may assume that the boundary condition L is of the form Lu = m(x′ ) Lν u + (m(x′ ) − 1)u,

(4.3) with

0 ≤ m(x′ ) ≤ 1

on ∂D.

Indeed, it suffices to note that the boundary condition Lu = m(x′ ) Lν u + γ(x′ ) (u|∂D ) = 0 on ∂D is equivalent to the condition

m(x′ ) m(x′ ) − γ(x′ )

Lν u +

γ(x′ ) m(x′ ) − γ(x′ )

(u|∂D ) = 0

on ∂D.

Furthermore, we remark that LG0α f = m(x′ ) Lν G0α f, and LHα ϕ = m(x′ ) Lν Hα ϕ + (m(x′ ) − 1)ϕ. Hence, in view of definition (3.13) it follows that (4.3′ )

Lu = m(x′ ) Lν u + (m(x′ ) − 1) (u|∂D ) ,

Therefore, the next theorem proves Theorem 2:

u ∈ D(L).

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63

Theorem 4.1. We define a linear operator W : C0 (D \ M ) −→ C0 (D \ M ) as follows (cf. formula (3.16)). (a) The domain D(W) of W is the set (4.4)

D(W) = {u ∈ C0 (D \ M ) : W u ∈ C0 (D \ M ),

Lu = m(x′ ) Lν u + (m(x′ ) − 1) (u|∂D ) = 0}.

(b) Wu = W u for all u ∈ D(W). Assume that the following condition (A′ ) is satisfied: (A′ ) 0 ≤ m(x′ ) ≤ 1 on ∂D. Then the operator W is the infinitesimal generator of some Feller semigroup {Tt }t≥0 on D \ M , and the Green operator Gα = (αI − W)−1 , α > 0, is given by the formula −1 (4.5) Gα f = Gνα f − Hα LHα (LGνα f ) , f ∈ C0 (D \ M ). Here Gνα is the Green operator for the boundary condition Lν given by formula (3.17): −1 ν 0 0 Lν Gα f , f ∈ C(D). Gα f = Gα f − Hα Lν Hα

Proof. We apply part (ii) of Theorem 1.3 to the operator W defined by formula (4.4), just as in the proof of Theorem 1. The proof is divided into several steps. (1) First, we prove that For all α > 0, the operator LHα generates a Feller semigroup on the boundary ∂D. 2−κ2 By virtue of the transmission property of T ∈ L1,0 (RN ), it follows (see [Bo], [RS, Chapter 3]) that the operator LHα is the sum of a degenerate elliptic differential operator of second order and a classical pseudo-differential operator of order 2 − min(κ1 , κ2 ):

LHα ϕ(x′ ) = m(x′ ) Lν Hα ϕ(x′ ) + (m(x′ ) − 1)ϕ(x′ ) N−1 N−1 2 X X ∂ ϕ ∂ϕ ′ i = m(x′ ) (x′ ) + (x ) αij (x′ ) β (x′ ) ∂xi ∂xj ∂xi i,j=1 i=1 + (m(x′ ) − 1) − αm(x′ ) δ(x′ ) ϕ(x′ ) ∂ (Hα ϕ) (x′ ) + m(x′ ) µ(x′ ) ∂n Z N−1 X ∂ϕ ′ ′ (yj − xj ) r(x′ , y ′ ) ϕ(y ′ ) − ϕ(x′ ) − + (x ) dy ∂x j ∂D j=1

64

KAZUAKI TAIRA

+

Z

D

t(x′ , y) Hα ϕ(y) − ϕ(x′ ) −

N−1 X j=1

(yj − xj )

∂ϕ ′ (x ) dy. ∂xj

Furthermore, it follows from an application of the boundary point lemma (see Appendix, Lemma A.2) that ∂ (Hα 1) < 0 on ∂D. ∂n This implies that LHα 1(x′ ) = m(x′ ) Lν Hα 1(x′ ) + (m(x′ ) − 1) ≤ 0

on ∂D.

Thus, by applying Theorem 2.1 to the operator LHα (cf. the proof of assertion (3.20)) we obtain that (4.6)

If λ > 0 is sufficiently large, then the range R(LHα − λI)

contains the space C 2+θ (∂D).

This implies that the range R(LHα −λI) is a dense subset of C(∂D). Therefore, by applying part (ii) of Theorem 3.14 to the operator L we obtain that the operator LHα is the infinitesimal generator of some Feller semigroup on ∂D, for all α > 0. (2) Now we prove that (4.7)

If condition (A′ ) is satisfied, then the equation LHα ψ = ϕ has a unique solution ψ in D LHα for any ϕ ∈ C(∂D); hence the inverse LHα

−1

of LHα can be defined on the whole space C(∂D).

Further the operator −LHα

−1

is non-negative and bounded on C(∂D).

Since we have, by inequality (3.19) and condition (A′ ), LHα 1(x′ ) = m(x′ )Lν Hα 1(x′ ) + (m(x′ ) − 1) ∂ = µ(x′ ) (Hα 1)(x′ ) + (m(x′ ) − 1) ∂nZ + m(x′ )

D

0. x′ ∈∂D

Here it should be noticed that the constants kα are increasing in α > 0: α ≥ β > 0 =⇒ kα ≥ kβ .

FELLER SEMIGROUPS AND MARKOV PROCESSES

65

Moreover, by using Corollary 1.4 with K := ∂D, A := LHα and c := kα we obtain that the operator LHα + kα I is the infinitesimal generator of some Feller semigroup on ∂D. Therefore, since kα > 0, it follows from an application of part (i) of Theorem 1.3 with A = LHα + kα I that the equation −LHα ψ = kα I − (LHα + kα I) ψ = ϕ

has a unique solution ψ ∈ D LHα for any ϕ ∈ C(∂D), and further that the −1 −1 is non-negative and bounded on the operator −LHα = kα I − (LHα + kα I) space C(∂D) with norm

−1 1 −1

.

−LHα = kα I − (LHα + kα I)

≤ kα

(4.8)

(3) By assertion (4.7), we can define the operator Gα by formula (4.5) for all α > 0. We prove that Gα = (αI − W)−1 ,

(4.9)

α > 0.

By virtue of Lemma 3.7 and Theorem 3.16, it follows that we have, for all f ∈ C0 (D \ M ), Gα f ∈ D(W ), and W Gα f = αGα f − f. Furthermore, we have (4.10)

LGα f =

LGνα f

− LHα LHα

−1

(LGνα f )

= 0 on ∂D.

However, we recall that (4.3′ )

Lu = m(x′ ) Lν u + (m(x′ ) − 1) (u|∂D ) ,

u ∈ D(L).

Hence we find that formula (4.10) is equivalent to the following: (4.10′ )

m(x′ ) Lν (Gα f ) + (m(x′ ) − 1) (Gα f |∂D ) = 0

on ∂D.

This implies that Gα f = 0

on M = {x′ ∈ ∂D : m(x′ ) = 0},

and so W Gα f = αGα f − f = 0 on M . Summing up, we have proved that Gα f ∈ D(W) = u ∈ C0 (D \ M ) : W u ∈ C0 (D \ M ), Lu = 0 ,

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and (αI − W)Gα f = f,

f ∈ C0 (D \ M ),

that is, (αI − W)Gα = I

on C0 (D \ M ).

Therefore, in order to prove formula (4.9) it suffices to show the injectivity of the operator αI − W for α > 0. Assume that u ∈ D(W) and (αI − W)u = 0. Then, by Corollary 3.8 it follows that the function u can be written in the form u|∂D ∈ D = D LHα .

u = Hα (u|∂D ) , Thus we have

LHα (u|∂D ) = Lu = 0. In view of assertion (4.7), this implies that u|∂D = 0, so that u = Hα (u|∂D ) = 0

in D.

(4) Now we prove the following three assertions: (i) The operator Gα is non-negative on the space C0 (D \ M ): f ∈ C0 (D \ M ), f ≥ 0

on D \ M =⇒ Gα f ≥ 0 on D \ M .

(ii) The operator Gα is bounded on the space C0 (D \ M ) with norm kGα k ≤

1 , α

α > 0.

(iii) The domain D(W) is everywhere dense in the space C0 (D \ M ). (i) First, we show the non-negativity of Gα on the space C(D): f ∈ C(D), f ≥ 0

on D =⇒ Gα f ≥ 0

on D.

Recall that the Dirichlet problem ′

(D )

(α − W )u = f u=ϕ

in D, on ∂D

is uniquely solvable. Hence it follows that (4.11)

Gνα f = Hα (Gνα f |∂D ) + G0α f

on D.

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67

Indeed, the both sides have the same boundary values Gνα f |∂D and satisfy the same equation: (α − W )u = f in D. Thus, by applying the operator L to the both sides of formula (4.11) we obtain that LGνα f = LHα (Gνα f |∂D ) + LG0α f. Since the operators −LHα

−1

and LG0α are non-negative, it follows that

−1 −1 ν ν 0 −LHα (LGα f ) = −Gα f |∂D + −LHα LGα f ≥ −Gνα f |∂D

on ∂D.

Therefore, by the non-negativity of Hα and G0α we find that −1 Gα f = Gνα f + Hα −LHα (LGνα f ) ≥ Gνα f − Hα (Gνα f |∂D ) = G0α f

≥ 0 on D. (ii) Next we prove the boundedness of Gα on the space C0 (D \ M ) with norm (4.12)

kGα k ≤

1 , α

α > 0.

To do this, it suffices to show that (4.12′ )

f ∈ C0 (D \ M ), f ≥ 0

on D =⇒ αGα f ≤ max f

on D,

D

since Gα is non-negative on the space C(D). We remark (cf. formula (4.3′ )) that LGνα f = m(x′ ) Lν Gνα f + (m(x′ ) − 1) (Gνα f |∂D ) = (m(x′ ) − 1) (Gνα f |∂D ) , so that (4.5′ )

−1 Gα f = Gνα f − Hα LHα (LGνα f ) −1 ν ′ ν = Gα f + Hα −LHα ((m(x ) − 1)Gα f |∂D ) . −1

Therefore, by the non-negativity of Hα and −LHα it follows that −1 Gα f = Gνα f + Hα −LHα ((m(x′ ) − 1)Gνα f |∂D ) ≤ Gνα f 1 ≤ max f α D

on D,

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since (m − 1)Gνα f |∂D ≤ 0 on ∂D and kGνα k ≤ 1/α. This proves assertion (4.12′ ) and hence assertion (4.12). (iii) Finally, we prove the density of D(W) in the space C0 (D \ M ). In view of formula (4.9), it suffices to show that (4.13)

lim kαGα f − f k∞ = 0,

α→+∞

f ∈ C0 (D \ M ) ∩ C ∞ (D).

We recall (cf. formula (4.5′ )) that (4.14)

−1 αGα f − f = αGνα f − f − αHα LHα (LGνα f ) −1 ν ′ ν = (αGα f − f ) + Hα LHα (α(1 − m(x ))Gα f |∂D ) .

We estimate each term on the right of formula (4.14). (iii-1) First, by applying Theorem 1 to the boundary condition Lν we find from assertion (3.32) that the first term on the right of formula (4.14) tends to zero: lim kαGνα f − f k∞ = 0.

(4.15)

α→+∞

(iii-2) To estimate the second term on the right of formula (4.14), we remark that −1 Hα LHα (α(1 − m(x′ ))Gνα f |∂D ) −1 −1 ′ ′ ν = Hα LHα ((1 − m(x ))f |∂D ) + Hα LHα ((1 − m(x ))(αGα f − f )|∂D ) .

However, we have, by assertion (4.8),

−1

(4.16)

Hα LHα ((1 − m(x′ )) (αGνα f − f ) |∂D ) ∞

−1

′ ν ≤ −LHα · k(1 − m(x )) (αGα f − f ) |∂D k∞ 1 k(1 − m(x′ )) (αGνα f − f ) |∂D k∞ kα 1 kαGνα f − f k∞ −→ 0 as α → +∞. ≤ k1 ≤

Here we have used the fact: k1 = − sup LH1 1(x′ ) ≤ kα = − sup LHα 1(x′ ) for all α ≥ 1. x′ ∈∂D

x′ ∈∂D

Thus we are reduced to the study of the term −1 Hα LHα ((1 − m(x′ ))f |∂D ) .

Now, for any given ε > 0, we can find a function h ∈ C ∞ (∂D) such that h = 0 near M = {x′ ∈ ∂D : m(x′ ) = 0}, k(1 − m(x′ ))f |∂D − hk∞ < ε.

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69

Then we have, for all α ≥ 1,

−1 −1

(4.17)

Hα LHα ((1 − m(x′ ))f |∂D ) − Hα LHα h ∞

−1

≤ −LHα · k(1 − m(x′ ))f |∂D − hk∞ ε ≤ kα ε ≤ . k1 Furthermore, we can find a function θ ∈ C0∞ (∂D) such that θ=1 near M , (1 − θ)h = h on ∂D. Then we have h(x′ ) = (1 − θ(x′ )) h(x′ ) 1 − θ(x′ ) ′ h(x′ ) = (−LHα 1(x )) −LHα 1(x′ ) 1 − θ(x′ ) ≤ sup khk∞ (−LHα 1(x′ )) . ′ x′ ∈∂D −LHα 1(x ) Since the operator −LHα

−1

is non-negative on the space C(∂D), it follows that 1 − θ(x′ ) −1 −LHα h ≤ sup · khk∞ on ∂D, ′ x′ ∈∂D −LHα 1(x )

so that (4.18)

−1

Hα LHα h

−1

≤ −LHα h

∞

∞

≤ sup

x′ ∈∂D

1 − θ(x′ ) −LHα 1(x′ )

However, there exists a constant c0 > 0 such that 0≤

1 − θ(x′ ) ≤ c0 , m(x′ )

x′ ∈ ∂D.

Hence we have 1 − θ(x′ ) m(x′ ) (−Lν Hα 1(x′ )) + (1 − m(x′ ))

1

≤ c0

−Lν Hα 1 .

1 − θ(x′ ) ≤ −LHα 1(x′ )

∞

In view of Lemma 3.19, this implies that 1 − θ(x′ ) = 0. lim sup α→+∞ x′ ∈∂D −LHα 1(x′ )

· khk∞ .

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Summing up, we obtain from inequalities (4.17) and (4.18) that

−1

lim sup Hα LHα ((1 − m(x′ ))f |∂D ) ∞ α→+∞ −1

≤ lim sup Hα LHα h ∞

α→+∞

−1 −1

′ + Hα LHα ((1 − m(x ))f |∂D ) − Hα LHα h ∞ ′ ε 1 − θ(x ) khk∞ + ≤ lim sup ′ α→+∞ x′ ∈∂D −LHα 1(x ) k1 ε ≤ . k1

Since ε is arbitrary, this proves that (4.19)

−1

lim Hα LHα ((1 − m(x′ ))f |∂D )

α→+∞

= 0.

∞

Therefore, by combining assertions (4.16) and (4.19) we find that the second term on the right of formula (4.14) also tends to zero:

−1

lim Hα LHα (α(1 − m(x′ ))Gνα f |∂D )

α→+∞

∞

= 0.

This completes the proof of assertion (4.13) and hence that of assertion (iii). (5) Summing up, we have proved that the operator W, defined by formula (4.4), satisfies conditions (a) through (d) in Theorem 1.3. Hence, in view of assertion (4.2) it follows from an application of part (ii) of the same theorem that the operator W is the infinitesimal generator of some Feller semigroup {Tt }t≥0 on D \ M . The proof of Theorem 4.1 and hence that of Theorem 2 is now complete. Appendix: The Maximum Principle In this appendix, following Bony–Courr`ege–Priouret [BCP] we formulate two useful maximum principles for second-order elliptic Waldenfels operators. First, we state the weak maximum principle. Theorem A.1 (The weak maximum principle). Let W be a second-order elliptic Waldenfels operator. Assume that a function u ∈ C(D) ∩ C 2 (D) satisfies either W u ≥ 0 and W 1 < 0 in D or W u > 0 and W 1 ≤ 0 in D. Then the function u may take its positive maximum only on the boundary ∂D. Next we consider the interior normal derivative (∂u)/(∂n) at a boundary point where the function u ∈ C 2 (D) takes its non-negative maximum. The boundary point lemma reads as follows:

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71

Lemma A.2 (The boundary point lemma). Let W be a second-order elliptic Waldenfels operator. Assume that a function u ∈ C(D) ∩ C 2 (D) satisfies W u ≥ 0 in D, and that there exists a point x′0 of ∂D such that

u(x′0 ) = maxx∈D u(x) ≥ 0,

u(x) < u(x′0 ),

x ∈ D.

Then the interior normal derivative (∂u)/(∂n)(x′0) of u at x′0 , if it exists, satisfies the condition ∂u ′ (x ) < 0. ∂n 0 For a proof of Theorem A.1 and Lemma A.2, the reader might refer to Bony– Courr`ege–Priouret [BCP] and also Taira [Ta1]. References [BCP] J.-M. Bony, P. Courr` ege et P. Priouret, Semi-groupes de Feller sur une vari´ et´ e ` a bord compacte et probl` emes aux limites int´ egro-diff´ erentiels du second ordre donnant lieu au principe du maximum, Ann. Inst. Fourier (Grenoble) 18 (1968), 369–521. [Bo] L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), 11–51. [Ca] C. Cancelier, Probl` emes aux limites pseudo-diff´ erentiels donnant lieu au principe du maximum, Comm. P.D.E. 11 (1986), 1677–1726. [CP] Chazarain, J. et A. Piriou, Introduction ` a la th´ eorie des ´ equations aux d´ eriv´ ees partielles lin´ eaires, Gauthier-Villars, Paris, 1981. [CM] R. R. Coifman et Y. Meyer, Au-del` a des op´ erateurs pseudo-diff´ erentiels, Ast´ erisque No. 57, Soc. Math. France, Paris, 1978. [Ho] L. H¨ ormander, The analysis of linear partial differential operators III, Springer-Verlag, Berlin Heidelberg New York Tokyo, 1985. [Ku] H. Kumano-go, Pseudo-differential operators, MIT Press, Cambridge, Mass., 1981. [OR] O.A. Ole˘ınik and E.V. Radkeviˇ c, Second order equations with nonnegative characteristic form, (in Russian), Itogi Nauki, Moscow, 1971; English translation, Amer. Math. Soc., Providence, Rhode Island and Plenum Press, New York, 1973. [RS] S. Rempel and B.-W. Schulze, Index theory of elliptic boundary problems, Akademie-Verlag, Berlin, 1982. [SU] K. Sato and T. Ueno, Multi-dimensional diffusion and the Markov process on the boundary, J. Math. Kyoto Univ. 14 (1965), 529–605. [Ta1] K. Taira, Diffusion processes and partial differential equations, Academic Press, San Diego New York London Tokyo, 1988. [Ta2] K. Taira, Boundary value problems and Markov processes, Lecture Notes in Math. No. 1499, Springer-Verlag, Berlin Heidelberg New York Tokyo, 1991. [Ta3] K. Taira, On the existence of Feller semigroups with boundary conditions, Memoirs Amer. Math. Soc. No. 475 (1992). [Ta4] K. Taira, On the existence of Feller semigroups with boundary conditions III, Hiroshima Math. J. 27 (1997), 77–103. [TW] S. Takanobu and S. Watanabe, On the existence and uniqueness of diffusion processes with Wentzell’s boundary conditions, J. Math. Kyoto Univ. 28 (1988), 71–80. [Ty] M. Taylor, Pseudodifferential operators, Princeton Univ. Press, Princeton, 1981. [Tr] H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. [Wa] S. Watanabe, Construction of diffusion processes with Wentzell’s boundary conditions by means of Poisson point processes of Brownian excursions, Probability Theory, Banach Center Publications, Vol. 5, PWN-Polish Scientific Publishers, Warsaw, 1979, pp. 255–271.

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[We]

[Yo]

A. D. Wentzell (Ventcel’), On boundary conditions for multidimensional diffusion processes (in Russian), Teoriya Veroyat. i ee Primen. 4 (1959), 172–185; English translation in Theory Prob. and its Appl. 4 (1959), 164–177. K. Yosida, Functional analysis, sixth edition, Springer-Verlag, Berlin Heidelberg New York, 1980.

Institute of Mathemati s, University of Tsukuba, Tsukuba 305{8571, Japan

E-mail address: [email protected]