Conservativeness and Extensions of Feller Semigroups - Springer Link

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REN É L. SCHILLING tions C. ∞ c (Rn ) ⊂ D(A), see Theorem 2.2—, we can express the above condition through continuity properties of the symbol of the ...
Positivity 2: 239–256, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

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Conservativeness and Extensions of Feller Semigroups RENÉ L. SCHILLING Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstraße 22–26 D-04103 Leipzig, Germany (Received: 23 January 1998; Accepted in revised form: 6 March 1998) Abstract. Let {Tt }t >0 denote a Feller semigroup on C∞ (Rn ), and {T˜t }t >0 its extension to the bounded measurable functions. We show that T˜t 1 ∈ Cb (Rn ) is necessary and sufficient for the invariance of Cb (Rn ) under {T˜t }t >0 . If the generator of the semigroup is a pseudo-differential operator we can restate this condition in terms of the symbol. As a by-product, we obtain necessary and sufficient conditions for the conservativeness of the semigroup which are again expressed through the symbol. Mathematics Subject Classifications (1991): 47D07, 60J35, 47G30. Key words: Feller semigroup; Feller process; conservativeness; positive maximum principle; pseudodifferential operator.

1. Introduction This paper deals with properties of canonical extensions of Feller semigroups and, in particular, their conservativeness. The starting point for this investigation was the observation that there are often different meanings attached to the word Feller. That a Feller semigroup consists of positivity preserving, sub-Markovian operators and that it is strongly continuous, is widely agreed on; disagreement, however, is about the space the operators act on. Usually, this is either the space of continuous functions vanishing at infinity, C∞ (R n ), with the topology of uniform convergence, or the space of bounded continuous functions, Cb (Rn ), equipped with locally uniform convergence. (Puzzling enough, there is no doubt that a strong Feller semigroup should map the bounded measurable functions Bb (Rn ) into Cb (R n ).) Here, we prefer the C∞ -version of the definiton, but we will prove a simple criterion that allows to extend Feller semigroups to Cb -Feller semigroups: if {T˜t }t >0 denotes the canonical extension of {Tt }t >0 onto Bb (Rn ), then T˜t 1 ∈ Cb (R n ) is necessary and sufficient that Cb (Rn ) is invariant w.r.t. T˜t , and {T˜t |Cb (R n )}t >0 is a Cb -Feller semigroup. If we know that the infinitesimal generator (A, D(A)) of {Tt }t >0 is (the extension of) a pseudo-differential operator—this is always the case if the test func-

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tions Cc∞ (Rn ) ⊂ D(A), see Theorem 2.2—, we can express the above condition through continuity properties of the symbol of the pseudo-differential operator, cf. Section 4. Clearly, if {Tt }t >0 is conservative, i.e., if T˜t 1 = 1, our criterion is (trivially) satisfied. It turns out that the methods leading to the above results can be modified so as to give (necessary and) sufficient criteria for the conservativeness of the semigroup. Roughly speaking, if the symbol of the generator satisfies supx∈Rn |p(x, ξ )| 6 c (1 + |ξ |2) (this should be interpreted as “the generator has uniformly bounded coefficients”), {Tt }t >0 will be conservative if and only if p(x, 0) ≡ 0. This result extends our earlier paper [10]. If we do not have “bounded coefficients”, then limk→∞ sup|y−x|62k sup|η|61/k |p(y, η)| = 0 still guarantees conservativeness. For the proof of the last criterion we only need the fact that the Feller process associated with the Feller semigroup satisfies the martingale problem with respect to the generator A|Cc∞ (R n ). It is, therefore, straightforward to generalize this result to stochastic processes that are (not a priori unique) solutions to a martingale problem for a given pseudo-differential operator. 2. Feller Semigroups and Pseudo-Differential Operators As we have already mentioned, there is no common agreement on how to define Feller semigroups. Here, we choose the following definition that is often used in probability theory, cf. [9, Chapter III.6]. DEFINITION 2.1. A Feller semigroup is a one-parameter semigroup {Tt }t >0 of positivity preserving sub-Markovian operators on the set of continuous functions vanishing at infinity—i.e., Tt : C∞ (Rn ) → C∞ (Rn ) and 0 6 Tt u 6 1 for all u ∈ C∞ (R n ) with 0 6 u 6 1—such that {Tt }t >0 is strongly continuous in the uniform (C∞ -) topology, lims→t kTs u − Tt uk∞ = 0 for all u ∈ C∞ (R n ) and t > 0. In Definition 2.1 we can replace the strong continuity of t 7 → Tt u, u ∈ C∞ (R n ), by the (usually encountered but equivalent) assumption that t 7 → Tt u is strongly continuous at t = 0 for u ∈ C∞ (Rn ) or by the even weaker condition limt →0 Tt u(x) = u(x) for all u ∈ C∞ (R n ) and x ∈ R n . The reason for the above redundancy is the fact that we want to study extensions of Feller semigroups. In the situation of Lemma 3.1 and Corollary 3.3, (strong) continuity at t = 0 will in general not be equivalent to strong continuity in the respective topologies. It is clear from Definition 2.1 that {Tt }t >0 is a (C0 )-contraction semigroup. The resolvent {Rλ }λ>0 associated with {Tt }t >0 is given by the (Bochner) integral Z ∞ Rλ u = e−λt Tt u dt, λ > 0, u ∈ C∞ (Rn ). (2.1) 0

It is easy to check that Rλ : C∞ (R n ) → C∞ (R n ) and that λRλ are positivity preserving, sub-Markovian operators satisfying the resolvent equation Rλ u − Rµ u = (µ − λ)Rµ Rλ u,

λ, µ > 0.

(2.2)

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Since (2.1) is a Laplace transform, there is a one-to-one correspondence between Feller semigroups and their resolvents. The infinitesimal generator (A, D(A)) of a Feller semigroup is given by Tt u − u t →0 t

Au := lim

(2.3)

 D(A) := u ∈ C∞ (R n ) : the limit (2.3) exists in C∞ (Rn ) ,

(2.4)

and one has the usual relations between semigroup, resolvent, and generator. Of particular importance will be the formula Z t Ts Au ds, u ∈ D(A), t > 0. (2.5) Tt u − u = 0

The generator satisfies the positive maximum principle, u ∈ D(A), u(x0 ) = sup u(x), u(x0 ) > 0 H⇒ Au(x0 ) 6 0, x∈Rn

(2.6)

as is immediately seen from the definition (2.3). The positive maximum principle allows deep insights into the structure of operators satisfying this property. The next theorem is due to Ph. Courrège, [3, Theorem 3.4] THEOREM 2.2. (Courrège). Let (A, D(A)) be a linear operator satisfying the positive maximum principle, Cc∞ (Rn ) ⊂ D(A), and A : C∞ (Rn ) → C(Rn ). Then A|Cc∞ (R n ) = −p(x, D) is a pseudo-differential operator, Z −n/2 (2.7) p(x, ξ )b u(ξ )eixξ dξ p(x, D)u(x) = (2π ) where the symbol p : Rn × R n → C is locally bounded and enjoys the following Lévy-Khinchine representation (as a function of ξ ) p(x, ξ ) =a(x) − i`(x) · ξ + ξ · Q(x)ξ Z  iyξ  1 − e−iyξ − N(x, dy), + 1 + |y|2 y6 =0

(2.8)

with measurable functions a(x) > 0, `(x) ∈ Rn , Q(x) = (qj k (x))nj,k=1 ∈ Rn×n symmetric, positive semidefinite, and a (measurable) kernel N(x, dy) on R n × R |y|2 B(Rn \ {0}) such that y6=0 1+|y| 2 N(x, dy) < ∞. Note that for every x ∈ R n the function ξ 7 → p(x, ξ ) is continuous negative definite (in the sense of A. Beurling), see [1] as standard reference. We will only need the following two properties of (continuous) negative definite functions: the subadditivity of the square root, p p p |p(x, ξ + η)| 6 |p(x, ξ )| + |p(x, η)|, ξ, η ∈ Rn , (2.9)

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and the growth behaviour, |p(x, ξ )| 6 2 sup |p(x, η)| (1 + |ξ |2 ),

6

|η| 1

ξ ∈ Rn .

(2.10)

It is not hard to see, cf. [11, Lemma 2.1, Remark 2.2] for a proof, that the symbol (x, ξ ) 7 → p(x, ξ ) is locally bounded if and only if for every compact set K ⊂ Rn one has   Z |y|2 sup a(x) + |`(x)| + |Q(x)| + N(x, dy) 6 cK < ∞. 2 x∈K y6 =0 1 + |y| (2.11) If we may admit K = R n , (2.11) will be equivalent to supx∈Rn |p(x, ξ )| 6 c (1 + |ξ |2 ). From the Lévy-Khinchine formula (2.8) we obtain with some elementary Fourier analysis the following integro-differential representation I (p) of −p(x, D), I (p)u(x) = − a(x)u(x) + `(x) · ∇u(x) + Z +

y6 =0



n X j,k=1

u(x − y) − u(x) +

qj k (x)

∂ 2 u(x) ∂xj ∂xk

y∇u(x)  N(x, dy), 1 + |y|2

for u ∈ Cc∞ (R n ). Since I (p) extends to Cb2 (R n ), we will view I (p) as extension of both −p(x, D) and A. Later on, we will frequently use the following elementary estimate. LEMMA 2.3. Let p(x, ξ ) be given by (2.8) and assumeRthat (x, ξ ) 7 → p(x, ξ ) is locally bounded. If g is a measurable function such that Rn (1 + |x|2 )|g(x)| dx < ∞, then Z sup |p(y, ξ/m)||g(ξ )| dξ 6 cg sup sup |p(y, η)|, m ∈ N , (2.12) y∈K |η|61/m Rn y∈K for all compact subsets K ⊂ Rn . Proof. If |ξ | > 1, we may write |ξ | = [|ξ |] + {|ξ |} where [|ξ |] ∈ N and {|ξ |} ∈ [0, 1). Using several times the subadditivity property (2.9) we find  ξ   1 ξ  ([|ξ |] + {|ξ |}) = p y, p y, m m |ξ |       6 2 p y, m1 |ξξ | [|ξ |] + p y, m1 |ξξ | {|ξ |}       6 2 [|ξ |]2 p y, m1 |ξξ | + p y, m1 |ξξ | {|ξ |}   6 2 (1 + |ξ |2 ) sup p y, mη . |η|61

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Consequently, Z  ξ  sup p y, |g(ξ )| dξ n m R y∈K  ξ  Z 6 sup sup p y, m |g(ξ )| dξ y∈K |ξ |61 |ξ |61 Z  η  + 2(1 + |ξ |2 ) sup sup p y, |g(ξ )| dξ m y∈K |η|61 |ξ |>1 Z 6 2 sup sup |p(y, η)| n (1 + |ξ |2) |g(ξ )| dξ, y∈K |η|61/m R



and the assertion follows.

3. Extending Feller Semigroups Let {Tt }t >0 be a Feller semigroup. By the Riesz representation theorem there exists a family of kernels pt (x, dy), t > 0, such that Z u(y) pt (x, dy), t > 0, u ∈ C∞ (R n ), (3.1) Tt u(x) = Rn holds true. Note that x 7 → pt (x, B) is measurable for every Borel set B ⊂ Rn and that B 7 → pt (x, B) is a sub-probability measure for every x ∈ R n . The right-hand side of (3.1) clearly extends to the bounded Borel measurable functions Bb (R n ) and defines a unique extension {T˜t }t >0 of the original semigroup {Tt }t >0. We observe that {T˜t }t >0 is again a semigroup of positive sub-Markovian operators, but lacks, in general, strong continuity. We will always use a tilde to denote this extension. As we have already mentioned, some authors prefer different definitions for Feller semigroups. Another quite common possibility is to replace (C∞ (Rn ), k•k∞ ) in Definition 2.1 by Cb (Rn ) equipped with local uniform convergence. For clarity’s sake, we will call this concept Cb -Feller semigroup. The next few results show which Feller semigroups extend to Cb -Feller semigroups. LEMMA 3.1. Let {Tt }t >0 be a Feller semigroup with the above extension {T˜t }t >0. Then we have limt →0 T˜t u = u locally uniformly for all u ∈ Cb (R n ). Proof. For u ∈ Cb (R n ) we set λ := λu := kuk∞ . We may assume that u > 0, otherwise we could consider positive and negative parts separately. Let K ⊂ R n be any compact set and choose χ ∈ Cc∞ (R n ) such that 1K 6 χ 6 1. Then (u − T˜t u) 1K = (χu − T˜t u) 1K

6 (χu − Tt (χu)) 1K

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  (T˜t u − u) 1K = (λ − u) − T˜t (λ − u) 1K + (T˜t λ − λ) 1K   6 (λ − u) − T˜t (λ − u) 1K

6 (χ(λ − u) − Tt (χ(λ − u))) 1K

where we applied the previous calculation to λ − u ∈ Cb (R n ). Thus, |u − T˜t u| 1K

6 |χu − Tt (χu)| + |χ(λ − u) − Tt (χ(λ − u))|

and the Lemma follows from the strong continuity of {Tt }t >0 since χu, χ(λ − u) ∈  C∞ (R n ). Denote by U SC(R n ) and LSC(Rn ) the sets of upper and lower semicontinuous functions on Rn . The subscript b stands for bounded functions. THEOREM 3.2. Let {Tt }t >0 be a Feller semigroup with resolvent {Rλ }λ>0 and denote their respective extensions to Bb (R n ) by {T˜t }t >0 and {R˜ λ }λ>0 . Then the following assertions are equivalent (i) (ii) (iii) (iv) (v)

>

T˜t 1 ∈ U SCb (Rn ), t 0; T˜t 1 ∈ Cb (Rn ), t 0; T˜t : Cb (Rn ) → Cb (Rn ), t 0; T˜t : U SCb (Rn ) → U SCb (Rn ), t T˜t : LSCb (Rn ) → LSCb (Rn ), t

>

>

> >

(vi) (vii) (viii) 0; (ix) 0; (x)

R˜ λ 1 ∈ U SCb (Rn ), λ > 0; R˜ λ 1 ∈ Cb (Rn ), λ > 0; R˜ λ : Cb (Rn ) → Cb (Rn ), λ > 0; R˜ λ : U SCb (Rn ) → U SCb (Rn ), λ > 0; R˜ λ : LSCb (Rn ) → LSCb (Rn ), λ > 0;

Proof. The proof of the equivalence of (vi)–(x) is almost identical with (i)–(v), so we will restrict ourselves to showing that (i)–(v) and (vii) are equivalent. Since U SCb (R n ) = −LSCb (R n ), and T˜t is linear, (iv) implies (v), and vice versa; but (iv)&(v) ⇒ (iii) ⇒ (ii) ⇒ (i) is obvious. (i) ⇒ (ii): Let {φk }k∈N be a sequence of functions φk ∈ Cc∞ (Rn ), 0 6 φk 6 1, increasing to 1. By monotone convergence T˜t 1 = T˜t (supk φk ) = supk Tt φk ∈ LSCb (Rn ) since Tt φk ∈ C∞ (Rn ) and Tt φk 6 1. This gives (ii). (ii) ⇒ (iii): Let u ∈ Cb (Rn ) and denote by {φk }k∈N the sequence used in the previous step. Since Tt φk increases towards T˜t 1, and since T˜t 1 is continuous, we conclude from Dini’s theorem, cf. [4, Theorem (7.2.2)], that this convergence is in fact locally uniform. Thus, |T˜t u − Tt (φk u)| = |T˜t ((1 − φk )u)| 6 kuk∞ (T˜t 1 − Tt φk ) → 0 locally uniformly as k → ∞. Since the Tt (φk u) are continuous, so is T˜t u. (iii) ⇒ (v): This is an easy consequence of monotone convergence, since every u ∈ LSCb (R n ) is the increasing limit of a sequence of continuous functions uk ∈ Cb (Rn ), i.e., T˜t u = supk Tt uk ∈ LSCb (R n ).

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R∞ (ii) ⇒ (vii): By monotone convergence we see R˜ λ 1 = 0 e−λt T˜t 1 dt. Since the integral converges absolutely, R˜ λ 1 ∈ Cb (R n ) if T˜t 1 ∈ Cb (R n ). (vii) ⇒ (ii): Assume that T˜t 1 is discontinuous at the point x. If {φk }k∈N is any sequence in C∞ (R n ) such that supk φk = 1, we have T˜t 1 = supk Tt φk , i.e., T˜t 1 ∈ LSCb (Rn ). Therefore, there exists a sequence {xk }k∈N , xk → x such that |x −xk | < 1 and lim sup T˜t 1(xk ) > lim inf T˜t 1(xk ) > T˜t 1(x). k→∞

k→∞

(3.2)

For given  > 0 we may assume that T˜t 1(xk ) > T˜t 1(x) + 2,

k ∈ N , k large.

(3.3)

By Lemma 3.1, limt →0 T˜t 1 = 1 locally uniformly, hence 1 6  + T˜s 1(y)

y ∈ B1 (x), s

for all

6 h,

(3.4)

with some small h = h > 0. Thus, T˜t 1(y) 6 T˜t +h1(y) + T˜t (y) 6 T˜s 1(y) +  for all y ∈ B1 (x) and s ∈ [t, t + h]. This, (3.4), and (3.3) yield in particular T˜s 1(x) 6 T˜t 1(x) 6 T˜t 1(xk ) − 2

6 T˜s 1(xk ) − 

for all s ∈ [t, t + h] and sufficiently large k ∈ N . By Fatou’s lemma we get Z ∞ ˜ e−λs lim inf T˜s 1(xk ) ds lim inf Rλ 1(xk ) > k→∞ k→∞ 0 Z t +h Z ∞ > e−λs T˜s 1(x) ds +  e−λs ds 0

> R˜ λ 1(x),

t

i.e., R˜ λ 1 is not continuous at x and we have reached a contradiction.



COROLLARY 3.3. Let {Tt }t >0 be a Feller semigroup such that T˜t 1 ∈ Cb (R n ). Then t 7 → T˜t u is for all u ∈ Cb (R n ) continuous with respect to locally uniform convergence. Proof. In view of Lemma 3.1 we can assume t > 0. For u ∈ Cb (R n ), T˜t u ∈ Cb (Rn ) by Theorem 3.2, and for h > 0 we know from Lemma 3.1, T˜t +h u = T˜h (T˜t u) → T˜t u locally uniformly as h → 0. On the other hand, we find some 0 < t0 < t, such that for a fixed compact set K ⊂ R n , χ ∈ Cc∞ (R n ) such that 1K 6 χ 6 1, and  > 0 T˜s (1 − χ) 6 

for all

06s

6 t − t0 , x ∈ K.

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Thus we have for all h 6 0, 0 < t0 < t + h < t and u ∈ Cb (R n ) |T˜t +h u − T˜t u| 1K   = T˜t +h−t0 T˜t0 u − T˜t0 −h u 1K     6 T˜t +h−t0 (1−χ)(T˜t0 u− T˜t0−h u) 1K + T˜t +h−t0 χ(T˜t0 u− T˜t0−h u) 1K

6 2kuk∞ T˜t +h−t (1 − χ) 1K + kχ(T˜t −hu − T˜t u)k∞ 6 2kuk∞  + kχ(T˜t +|h| u − T˜t u)k∞ . 0

0

0

0

0

Since  > 0 is arbitrarily small and since the second term vanishes as h → 0 by the first part of the proof, the assertion follows.  COROLLARY 3.4. Every Feller semigroup {Tt }t >0 satisfying T˜t 1 ∈ Cb (Rn ) extends to a Cb -Feller semigroup. The converse question, whether a Cb -Feller semigroup preserves the set C∞ (R n ) seems to have no general answer. Of course, if its generator A maps Cc∞ (Rn ) into C∞ (R n ), one could appeal to the Hille-Yosida-Ray theorem in order to show that A actually generates a Feller semigroup in the sense of Definition 2.1. Conditions of this type were, in a different context, studied by W. Hoh, ([6], Theorem 3.3), see also Remark 4.5 below.

4. Conditions Ensuring the Continuity of T˜t 1 The preceding section dealt with rather general Feller semigroups. In order to give nice conditions ensuring T˜t 1 ∈ Cb (R n ), we have to make assumptions on the richness of the domain D(A) of the generator. Although some abstract condition of the type “there is a sequence {φk }k∈N ⊂ D(A) such that 1Bk (0) 6 φk 6 1B2k (0) and supk kAφk k∞ < ∞” will often do, we prefer to assume that Cc∞ (Rn ) ⊂ D(A) which implies A|Cc∞ (R n ) = −p(x, D), see Theorem 2.2. LEMMA 4.1. Let {Tt }t >0 be a Feller semigroup with generator (A, D(A)) such that Cc∞ (Rn ) ⊂ D(A) and A|Cc∞ (Rn ) = −p(x, D) with a locally bounded symbol p(x, ξ ) given by (2.8). Then for every φ ∈ Cc∞ (R n ) satisfying 0 6 φ 6 1 and φ(0) = 1, the sequence φk (x) := φ( xk ), k ∈ N , tends to 1 and p(x, D)φk (x) → p(x, 0) as k → ∞. If supx∈Rn |p(x, ξ )| 6 c (1 + |ξ |2 ), one has supk∈N supx∈Rn |p(x, D)φk (x)| < ∞.

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φ (kξ ). Therefore, Proof. Clearly, φbk (ξ ) = k n b Z −n/2 p(x, ξ )φbk (ξ )eixξ dξ |p(x, D)φk (x)| = (2π ) n ZR  ξ −n/2 = (2π ) p x, b φ (ξ )eixξ/k dξ n k Z R  ξ −n/2 6 (2π ) φ (ξ ) dξ p x, b n k ZR ξ 2   −n/2 6 (2π ) 2 sup |p(x, η)| 1 + |b φ (ξ )| dξ n k R |η|61 Z 6 2(2π )−n/2 sup |p(x, η)| n (1 + |ξ |2 )|bφ (ξ )| dξ. |η|61 R

(4.1) (4.2)

This calculation shows that we may use dominated convergence in (4.1), (4.2) to deduce that limk→∞ p(x, D)φk (x) = p(x, 0) for every x ∈ R n . If supx∈Rn |p(x, ξ )| 6 c (1 + |ξ |2), it shows also the uniform boundedness of p(x, D)φk (x) in k ∈ N and x ∈ Rn .  REMARK 4.2. (i) Note that we can choose the function φ in the above lemma such that {x : φk (x) = 1} ↑ R n as k → ∞ and φk increases to 1. For example, take f ∈ Cc∞ (R), 1B1 (0) 6 f 6 1B2 (0) and decreasing if 1 6 |r| 6 2 and set φ(x) := f (|x|), x ∈ Rn . (ii) If (−p(x, D), Cc∞ (R n )) satisfies the positive maximum principle and maps into C(Rn ), the function x 7 → p(x, 0) is upper semicontinuous. To see this, choose the sequence {φk }k∈N as above and fix any compact set K ⊂ R n . For all j, k ∈ N , j > k > k0 and K ⊂ Bk0 (0) we have because of the positive maximum principle Cc∞ (Rn )

p(x, D)(φj − φk ) = −p(x, D)(φk − φj ) 6 0 for all

x ∈ K,

and by Lemma 4.1, infk>k0 1K (x)p(x, D)φk (x) = p(x, 0), i.e., it is in the interior of K the lower envelope of a sequence of continuous functions. Since K was arbitrary, p(x, 0) is upper semicontinuous. We can now state the criterion for T˜t 1 ∈ Cb (Rn ). THEOREM 4.3. Let {Tt }t >0 be a Feller semigroup with generator (A, D(A)) such that Cc∞ (Rn ) ⊂ D(A) and A|Cc∞ (Rn ) = −p(x, D) with symbol p(x, ξ ) given by (2.8). Assume, moreover, that supx∈Rn |p(x, ξ )| 6 c (1 + |ξ |2 ). If x 7 → p(x, 0) is continuous, so is x 7 → T˜t 1(x). Proof. Let {φk }k∈N be the sequence constructed in Remark 4.2. Then we find by monotone and dominated convergence and the uniform boundedness of Aφk (x) in k and x Z t Z t T˜s p(•, 0) ds. Ts p(•, D)φk ds = 1 − T˜t 1 = lim (φk − Tt φk ) = lim k→∞

k→∞

0

0

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Since p(•, 0) ∈ Cb (Rn ), we have in particular p(•, 0) ∈ LSCb (R n ). Thus, there is a sequence uk ∈ Cc (R n ) increasing towards p(•, 0), and by monotone convergence Z t Z t Z t ˜ ˜ Ts p(•, 0) ds = Ts (sup uk ) ds = sup Ts uk ds ∈ LSCb (R n ), 0

0

Rt

k

k

0

because 0 Ts uk ds ∈ Cb (Rn ) and T˜s p(•, 0) is bounded. A similar argument shows that T˜t 1 = supk Tt φk ∈ LSCb (Rn ), hence Z t n ˜ T˜s p(•, 0) ds ∈ LSCb (R n ), U SCb (R ) 3 1 − Tt 1 = 0



and therefore T˜t 1 ∈ Cb (R n ). Our next theorem states several equivalent conditions for p(•, 0) ∈ Cb (Rn ).

THEOREM 4.4. Let −p(x, D) be a pseudo-differential operator satisfying the positive maximum principle (2.6) on Cc∞ (R n ) and with the additional property that p(x, D) maps Cc∞ (R n ) into C(R n ). Then the symbol p(x, ξ ) is given by (2.8), locally bounded, and the following assertions are equivalent: (i)

x 7 → p(x, 0) ∈ C(Rn );

(ii) lim|ξ |→0 supx∈K |p(x, ξ ) − p(x, 0)| = 0 for all compact sets K ⊂ R n ; (iii) limR→∞ supx∈K N(x, BRc (0)) = 0 for all compact sets K ⊂ Rn ; (iv) x 7 → p(x, ξ ) ∈ C(R n ) for all ξ ∈ R n . Proof. We begin with (ii) ⇔ (iii), and prove then (iii) ⇒ (iv) ⇒ (i) ⇒ (iii). Throughout the proof {φk }k∈N denotes the sequence of functions increasing to 1 that was constructed in Remark 4.2. (iii) ⇒ (ii): Fix some compact set K ⊂ R n . Since (x, ξ ) 7 → p(x, ξ ) is locally bounded, we know from (2.11) and the remark preceding it that supx∈K (|`(x)| + |Q(x)|) < ∞, thus lim sup(|`(x) · ξ | + |ξ · Q(x)ξ |) = 0,

|ξ |→0 x∈K

and we may assume that ` ≡ 0 and Q ≡ 0. Therefore, Z  iyξ  |p(x, ξ ) − p(x, 0)| = 1 − e−iyξ − N(x, dy) 2 1 + |y| y6 =0 Z iyξ −iyξ 6 − 1 − e N(x, dy) 1 + |y|2 0R Z iyξ 6 1 − e−iyξ − N(x, dy) + (2 + |ξ |) N(x, BRc (0)). 1 + |y|2 0 1  η  p x, 2 − p(x, 0) R Z 2 η  |y|2  1 2 η 6 ) + |y| (1 + |y| 2 N(x, dy) 2 2 2 1 + |y| R R 0R 1 + |y/R| Z  Z ηy  1 − cos = g(η) dη N(x, dy) R |y|>R Rn Z     6 n Re p x, Rη − p(x, 0) g(η) dη R where the function g is given by Z 1 ∞ 2 (2π r)−n/2 e−|η| /(2r) e−r/2 dr. g(η) := 2 0 Note that g has moments of arbitrary order. Using Lemma 2.3 we get   η  N(x, BRc (0)) 6 cg sup Re p x, − p(x, 0) R |η|61

6 cg

sup (Re p(x, ξ ) − p(x, 0)),

6

|ξ | 1/R

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and for all compact sets K ⊂ R n lim sup N(x, BRc (0)) 6 2cg lim

R→∞ x∈K

sup sup (Re p(x, ξ ) − p(x, 0))

6

R→∞ |ξ | 1/R x∈K

= 2cg lim sup sup (Re p(x, ξ ) − p(x, 0)) = 0. |ξ |→0 x∈K

(iii) ⇒ (iv): We set fk (x) := eixξ φk (x). Using essentially the same arguments as in the proof of Lemma 4.1, we can use dominated convergence in order to find lim e−ixξ p(x, D)fk (x) = p(x, ξ ),

k→∞

x, ξ ∈ R n .

We are going to show that this convergence is locally uniform in x. Fix a compact subset K ⊂ R n and choose j > k such that K ⊂ Bk/2 (0). For x ∈ K −ixξ p(x, D)fj (x) − e−ixξ p(x, D)fk (x) e Z = (fj (x − y) − fk (x − y)) N(x, dy) y6 =0 Z 6 (φj (x − y) − φk (x − y)) N(x, dy). y6 =0

Let k0 = sup{|x| : x ∈ K}. Now K ⊂ Bk0 (0) ⊂ Bk/2 (0) and for x ∈ K |y| > 2j + k0 |y| < k − k0

implies |x − y| > |y| − |x| > 2j implies |x − y| 6 |y| + |x| < k.

In both cases φj (x − y) − φk (x − y) = 0, therefore 1K (x) e−ixξ p(x, D)fj (x) − e−ixξ p(x, D)fk (x) Z 6 1K (x) (φj (x − y) − φk (x − y)) N(x, dy)

6

6 6

k−k0 |y| 2j +k0 c 1K (x) N(x, Bk−k (0)) 0

and the latter tends to 0 uniformly as k → ∞ (and, a fortiori, j → ∞). (iv) ⇒ (i): Obvious. (i) ⇒ (iii): Fix some compact set K ⊂ Rn and choose k0 such that K ⊂ Bk0 (0). For all j > k > 2k0 we find from the positive maximum principle that 1K (x) p(x, D)(φk − φj )(x) > 0, and therefore 1K (x) p(x, 0) = limk→∞ 1K (x) p(x, D)φk+k0 (x) is a decreasing limit of continuous functions. Since p(x, 0) is itself continuous and K an arbitrary compact set, p(x, D)φk (x) → p(x, 0) locally uniformly by Dini’s theorem.

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Exactly as in (iii) ⇒ (iv), we find 1K (x) |p(x, D)φk (x) − p(x, D)φj (x)| Z = 1K (x) (φj (x − y) − φk (x − y)) N(x, dy)

6 6

k−k0 |y| 2j +k0

which yields as j → ∞

Z

1K (x) |p(x, D)φk (x) − p(x, 0)| = 1K (x)

>

|y| k−k0

(1 − φk (x − y)) N(x, dy).

Since for x ∈ K φk (x − y) 6 1B2k (0) (x − y) = 1B2k (0)+x (y) 6 1B2k (0)+K (y) 6 1B2k+k0 (0) (y), we have

Z

1K (x) |p(x, D)φk (x)−p(x, 0)| = 1K (x)

> Z

|y| k−k0

(1−φk (x −y)) N(x, dy)

> 1K (x) (1−1B |y|>k−k c > 1K (x) N(x, B2k+k (0)).

2k+k0 (0)

(y)) N(x, dy)

0

0

As p(x, D)φk (x) → p(x, 0) locally uniformly, we get lim sup N(x, BRc (0)) 6 lim sup |p(x, D)φk (x) − p(x, 0)| = 0.

R→∞ x∈K

k→∞ x∈K

 REMARK 4.5. (i) The equivalence of (ii) and (iii) in the preceding theorem was— for K = Rn and jointly continuous symbols—remarked by W. Hoh, [6, Theorem 3.3]. Hoh used it in order to show that a pseudo-differential operator q(x, D) with jointly continuous symbol q(x, ξ ) maps Cc∞ (Rn ) into C∞ (Rn ). (ii) It is not hard to see that in the situation of Theorem 4.4 x 7 → p(x, D)1(x) ∈ C(Rn ) if and only if

p(x, D) : Cb2 (R n ) → C(R n ).

The necessity is trivial, for the sufficiency we observe that for any compact subset K ⊂ Rn , all f ∈ Cb (R n ), the sequence {φk }k∈N of Remark 4.2, and sufficiently large k we have 1K (x)|p(x, D)f (x) − p(x, D)(φk f )(x)| Z = 1K (x) f (x − y)(1 − φk (x − y)) N(x, dy) y6 =0 Z 6 1K (x)kf k∞ (1 − φk (x − y)) N(x, dy) → 0 y6 =0

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locally uniformly as k → ∞. (iii) The, maybe, most astonishing fact is that (i) and (iv) are equivalent conditions. This shows, in particular, that the counterexample of Courrège (see below) of an operator A : Cc∞ (R) → C(R ) satisfying the positive maximum principle and such that A1 is discontinuous, is generic. EXAMPLE 4.6. (Courrège [3, Section 3.3], Bony, Courrège, Priouret [2, Remark I.2.10]) Set N(x, dy) = δx− 1 (dy) if x 6 ∈ {−1, 0, 1} and = 0 otherwise. Then x

Z Au(x) : =

y6 =0

( =

(u(x − y) − u(x)) N(x, dy) − 1{0} (x)u(x)

−u(0),

if x = 0

u( x1 ) − u(x), if x 6 = 0

It is obvious that A maps the test functions Cc∞ (R) into the continuous functions, and if u attains at x0 its positive maximum, then Au(x0 ) 6 0. The corresponding symbol is given by p(x, ξ ) = −e−ixξ A(ei •ξ )(x) = 1{0} (x) + 1R\{0} (x)(1 − ei( x −x)ξ ). 1

Note that this operator does not generate a Feller semigroup.

5. Conservativeness of Feller Semigroups A Feller semigroup {Tt }t >0 is said to be conservative, if T˜t 1 = 1 for all t > 0. Note that in this case trivially T˜t 1 ∈ Cb (R n ), hence the material of the previous sections is applicable. Under certain restrictions—typically we assumed that A|Cc∞ (Rn ) = −p(x, D) and that the symbol p(x, ξ ) was bounded and differentiable in x—we showed in [10] that a Feller semigroup is conservative if and only if p(x, 0) ≡ 0. We will both simplify this proof and extend it to a much larger class of Feller semigroups. Roughly speaking, we will only assume that A is the extension of a pseudo-differential operator −p(x, D). Before giving the proof, we want to explain conservativeness from a probabilistic point of view. Let (, A, P, {Px }x∈Rn , {Xt }t >0, {Ft }t >0, R n , B(R n )) be the stochastic (Feller) process associated with the Feller semigroup {Tt }t >0. It is wellknown that the transition probabilities of {Xt }t >0 are just the kernels pt (x, dy) used in (3.1) to extend Tt . The random time ζ(ω) := inf{t

> 0 : Xt 6∈ Rn },

ω ∈ ,

(5.1)

is called lifetime of the process because it is the instant the process ceases to be finitely valued. Probabilists kill the process once its clock has reached ζ(ω). This

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253

can be accomplished by a one-point compactification of the state space R n , say R n1 . One sets pt (1, {1}) := 1 and

pt (1, R n ) := 0,

i.e., 1 is the point of no return. The important observation is now that P• (ζ = ∞) ≡ 1

if and only if T˜t 1 ≡ 1, t

> 0,

(note that still 1 = 1Rn ). This is easily seen because

1 > T˜t 1(x) = E x (1 ◦ Xt ) = Px (Xt ∈ Rn ) = Px (ζ > t) > Px (ζ = ∞)

and Px (ζ = ∞) = lim Px (ζ > t) = lim T˜t 1(x). t →∞

t →∞

Let us first treat the case where the symbol p(x, ξ ) has “uniformly bounded coefficients”. LEMMA 5.1. Let {Tt }t >0 be a Feller semigroup with generator (A, D(A)) such that Cc∞ (R n ) ⊂ D(A) and A|Cc∞ (R n ) = −p(x, D) with symbol p(x, ξ ) given by (2.8) satisfying p(•, 0) ∈ C(R n ) and supx∈Rn |p(x, ξ )| 6 c (1 + |ξ |2 ). Then limt →0 1t (1(x) − T˜ 1(x)) = p(x, 0) for every x ∈ R n . Proof. Let {φk }k∈N be the sequence constructed in Remark 4.2, φk ↑ 1. By monotone and dominated convergence we find as in the proof of Theorem 4.3 Z 1 t ˜ 1 ˜ Ts p(•, 0) ds, t > 0. (1 − Tt 1) = t t 0 Since p(•, 0) ∈ CRb (Rn ), Lemma 4.1 shows that lims→0 T˜s p(•, 0) = 0. This, in turn, t  implies limt →0 1t 0 T˜s p(•, 0) ds = p(•, 0), and the assertion follows. We can now state the main theorem generalizing our results in [10]. THEOREM 5.2. Let {Tt }t >0 be a Feller semigroup with generator (A, D(A)) such that Cc∞ (R n ) ⊂ D(A) and A|Cc∞ (R n ) = −p(x, D) with symbol p(x, ξ ) given by (2.8) satisfying supx∈Rn |p(x, ξ )| 6 c (1 + |ξ |2 ). The semigroup {Tt }t >0 is conservative, if p(x, 0) ≡ 0. Conversely, if {Tt }t >0 is conservative and p(•, 0) continuous, then p(x, 0) ≡ 0. Proof. If p(x, 0) ≡ 0, we have p(•, 0) ∈ Cb (Rn ) and by Theorem 4.3 T˜t 1 ∈ Cb (Rn ). Now Z t 1 − T˜t 1 = T˜s p(•, 0) ds = 0, 0

thus T˜t 1 = 1. Conversely, assume that T˜t 1 = 1. By our assumption, p(•, 0) ∈ Cb (R n ), and we find with Lemma 5.1 for every x ∈ R n that limt →0 1t (1(x) − T˜t 1(x)) = p(x, 0).  This shows that p(x, 0) = 0.

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REMARK 5.3. If in Theorem 5.2 p(x, D) is a local operator, which is to say that N(x, dy) vanishes, then p(•, 0) is necessarily continuous. Indeed, if K ⊂ R n is some compact set and χK ∈ Cc∞ (R n ) such that 1K 6 χK 6 1, the local property implies p(x, 0) = p(x, D)1(x) = p(x, D)χK (x),



x ∈K ,

and p(•, D)χK ∈ C(R n ) because of the mapping properties of p(x, D). For nonlocal operators, this seems to be an open question, see, however, Example 4.6. COROLLARY 5.4. In the setting of Theorem 5.2 denote by {Xt }t >0 the Feller process associated with {Tt }t >0 and assume p(•, 0) ∈ Cb (Rn ). Set λt (x, ξ ) := E x (ei(x−Xt )ξ ). Then d λt (x, ξ ) = −p(x, ξ ). t =0 dt Proof. Replacing φk in the proof of Lemma 5.1 by fk (x) := eixξ φk (x)—just as in Theorem 4.4, (iii) ⇒ (iv)—and some obvious modifications yield the Corollary.



The significance of Corollary 5.4 is that it allows to identify the symbol via stochastic considerations. The function λt (x, ξ ) was introduced by Jacob [7], where he pointed out that it should be interpreted as characteristic function (in the language of probability theory) of the process {Xt }t >0. The shift in the argument is necessary in order to take into account the (spatially) non-homogeneous nature of the process. Theorem 5.2 is only of limited use if one does not know about the Feller property of {Tt }t >0 or {Xt }t >0. This is, e.g., the case when one constructs a stochastic process by solving the martingale problem (cf. [5] for precise definitions) for a given operator A satisfying the positive maximum principle and A : Cc∞ (Rn ) → C(R n ): here one looks for a process {Xt }t >0 such that Z t Au(Xs ) ds is a local martingale for all u ∈ Cc∞ (R n ) (5.2) u(Xt ) − 0

where the underlying filtration is the natural filtration σ (Xs , s 6 t). Note that by Theorem 2.2 A|Cc∞ (Rn ) is necessarily a pseudo-differential operator −p(x, D) with p(x, ξ ) as in (2.8). Now {Xt }t >0 is some solution to (5.2), and there might be different ones. In order to show uniqueness (thus, well-posedness of the martingale problem), it is often helpful to have criteria for the conservativeness of the solution. This is what the next theorem is about. In order to see that it is also a generalization of Theorem 5.2 to unbounded “coefficients”, note that every Feller process with generator (A, D(A)) and Cc∞ (Rn ) ⊂ D(A) automatically satisfies (5.2), see [5, Proposition 4.1.7]. THEOREM 5.5. Let {Xt }t >0 be a stochastic process with càdlàg paths satisfying (5.2) with A = −p(x, D). Then {Xt }t >0 is conservative (i.e., has almost surely infinite lifetime) if limk→∞ sup|y−x|62k sup|η|61/k |p(y, η)| = 0 for all x ∈ R n .

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Proof. Let {φk }k∈N be the sequence of functions φk ↑ 1 constructed in Remark b(kξ ), and |φb1x | = |φb10 | = 4.2 and set φkx (y) := φk (y −x). Clearly, φbkx (ξ ) = e−ixξ k n φ b |φ |. We denote by τk (ω) := τkx (ω) := inf{t > 0 : |Xt − x| > k} the first time Xt leaves the ball Bk (x). Observe that τk 6 ζ and φkx (Xτ2k ) = 0 since supp φkx ⊂ B2k (x). Thus, the stopped process Xtτ2k (ω) := Xmin{t,τ2k (ω)} (ω) satisfies (

1 ◦ Xt = 0 1 ◦ Xt = 1

> >

φkx (Xτ2k ) = φkx (Xtτ2k ), if t φkx (Xtτ2k ),

> ζ (> τ2k );

if t < ζ ;

Applying a standard optional stopping argument shows 1 − E x (1 ◦ Xt ) 6 1 − E x (φkx (Xtτ2k ))  Z min{t,τ2k }  x x 6E p(y, Dy )φk (y) y=Xs ds 0

Z = Ex

min{t,τ2k }

0

 x )φ (y) D ds p(y, y k y=Xs−

since Xs has at most countable many jumps on [0, t], i.e., {s ∈ [0, t] : Xs (ω) 6 = Xs− (ω)} is for every ω ∈  a set of Lebesgue measure zero. Plugging in the definition of p(y, D)φkx (y) gives  Z iXs− ξ bx 1 − E (1 ◦ Xt ) 6 (2π ) φk (ξ ) dξ ds E p(Xs−, ξ )e n R 0  Z min{t,τ2k } Z   ξ b 6 (2π )−n/2 E x , p X φ (ξ ) dξ ds s− k Rn 0  Z    6 (2π )−n/2 E x min{t, τ2k } n sup p Xs−, ξk φb(ξ ) dξ R s 6min{t,τ2k } Z  ξ 6 (2π )−n/2 t n sup p y, k φb(ξ ) dξ R |y−x|62k 6 cφ t sup sup |p(y, η)| x

−n/2

Z

x

6

min{t,τ2k }

6

|y−x| 2k |η| 1/k

where we used Lemma 4.1 for the ultimate step. Thus Px (Xt 6 ∈ Rn ) = 1 − E x (1 ◦ Xt )

6 cφ t

sup

6

sup |p(y, η)| → 0

6

|y−x| 2k |η| 1/k

as k → ∞, i.e., supt >0 Px (Xt 6 ∈ R n ) = 0 which shows that Px (ζ = ∞) = 1.



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Acknowledgements I would like to thank Dr. W. Hoh, Bielefeld, for several discussions on the case of unbounded symbols treated in Section 5 below. Independently of the result presented here, W. Hoh has proved a similar criterion for the conservativeness of stochastic processes in terms of their symbols. His result will be contained in his forthcoming Habilitationsschrift. Financial support by DFG post-doctoral fellowship Schi 419/1–1 is gratefully acknowledged. References 1. 2.

3. 4. 5. 6. 7. 8. 9. 10. 11.

Berg, C. and G. Forst, Potential Theory on Locally Compact Abelian Groups, Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete, II. Ser. Bd. 87, Berlin 1975. Bony, J.-M., Courrège, Ph. et P. Priouret, Semi-groupes de Feller sur une variété à bord compacte et problème aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum, Ann. Inst. Fourier, Grenoble 18.2 (1968), 369–521. ∞ dans C satisfaisant au Courrège, Ph., Sur la forme intégro-différentielle des opérateurs de CK principe du maximum, Sém. Théorie du Potentiel (1965/66) 38 p. Dieudonné J., Foundations of Modern Analysis, Academic Press, International Edition, New York 1969 (enlarged and corrected printing). Ethier, St. E. and Th. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, Series in Probab. and Math. Stat., New York 1986. Hoh, W., On perturbations of pseudo differential operators with negative definite symbol, preprint (1997). Jacob, N., Characteristic functions and symbols in the theory of Feller processes, Potential Analysis 8 (1998), 61–68. Jacob, N., Pseudo-differential operators and Markov processes, Akademie Verlag, Mathematical Research vol. 94, Berlin 1996. Rogers, L. C. G. and D. Williams, Diffusions, Markov Processes, and Martingales. Volume one: Foundations, Wiley, Series in Probab. Math. Stat., Chichester 1994 (2nd ed.). Schilling, R. L., Conservativeness of semigroups generated by pseudo differential operators, to appear in Potential Analysis. Schilling, R. L., Growth and Hölder conditions for the sample paths of Feller processes, to appear in Probab. Theory Relat. Fields.

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