MARKOV TRANSITION FUNCTIONS AND SEMIGROUPS OF

MARKOV TRANSITION FUNCTIONS AND SEMIGROUPS OF MEASURES TIMOTHY LANT† AND HORST R. THIEME¦ Abstract. The application of operator semigroups to Markov processes is extended to Markov transition functions which do not have the Feller property. Markov transition functions are characterized as solutions of forward and backward equations which involve the generators of integrated semigroups and are shown to induce integral semigroups on spaces of measures.

1. Introduction Continuous semigroups of bounded linear operators have played an important role as functional analytic tools in the theory of Markov processes [15, 16, 26]. The link is provided by (Markov) transition functions K(t, x, D). Here x is a point in a state space Ω and D an element of a σ-algebra B of subsets of Ω; t ≥ 0 is interpreted as time and K(t, x, D) as the probability of the state being in the set D at time t provided x was the state at time 0. There are at least two ways of associating operator families with K (Section 2). The first type of family operates on the Banach space of (signed) measures on B with bounded variation, M(Ω), and is defined by Z (1.1) [S(t)µ](D) = µ(dx)K(t, x, D), µ ∈ M(Ω), Ω

[16, X.8], the second is the formally dual family on the Banach space of bounded measurable functions on Ω, BM(Ω), with supremum norm, Z ˜ (1.2) [S(t)f ](x) = K(t, x, dy)f (y), f ∈ BM(Ω), Ω

Date: May 10, 2006. 1991 Mathematics Subject Classification. 47D06, 47D62, 60J35. Key words and phrases. (Markov) transition functions, stochastic continuity, semigroups (C0 -, integrated, integral), forward and backward equations, Feller property, spaces of measures. † partially supported by NSF grants DMS-0314529 and SES-0345945. ¦ partially supported by NSF grants DMS-9706787 and DMS-0314529. 1

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T. Lant, H.R. Thieme

[19, Sec.1.9]. The space of measures has an important interpretation in another application of Markov transition functions, structured population models, as a positive measure µ represents the population distribution with respect to the structure induced by the individual state space Ω [9]. In this context, S(t)µ represents the structural distribution ˜ are oneof the population at time t. Both operator families, S and S, parameter semigroups. The semigroup property, S(t)S(r) = S(t + r) for all t, r ≥ 0, is equivalent to the Chapman-Kolmogorov equation Z (1.3) K(t + r, x, D) = K(r, x, dy)K(t, y, D) ∀t, r ≥ 0. Ω

Among the many types of operator semigroups [21], C0 -semigroups [6, 7, 14, 19, 25, 31] are the most studied. Unfortunately, neither S(t)µ nor ˜ S(t)f are continuous in t in general. However, meaningful conditions ˜ concerning K can be found for S(t)f to be continuous in t if f is a continuous function on a suitable Hausdorff topological space, f ∈ C(Ω). Let Ω be a compact metric space for the sake of exposition (we will also look at the cases where Ω is a normal space or a locally compact ˜ space). The continuity of S(t)f in t for f ∈ C(Ω) can most easily be ˜ leaves C(Ω) invariant such that exploited in a semigroup setting if S(t) ˜ the restrictions of S(t) to C(Ω) form a C0 -semigroup. Historically, this property, called the Feller property [8], has flatly been postulated. The associated transition function K is then called a Feller function [26, p.56]. Feller functions are measure-theoretically characterized as follows (Sections 4 and A, cf. [31]): (•) If D is an open set which is the countable union of compact sets (an open Kσ set) and t ≥ 0, then K(t, x, D) is a lower semi-continuous function of x ∈ Ω. If D is a compact set which is the countable intersection of open sets (a compact Gδ set), then K(t, x, D) is an upper semi-continuous function of x ∈ Ω. For K to induce a C0 -semigroup on C(Ω) it is necessary and sufficient to have the respective semi-continuity properties (•) to hold in (t, x) ∈ R+ × Ω (Sections 4 and A). If they hold and Ω is a compact metric space, S is a dual semigroup on M(Ω) [5, 6, 28, 32] as M(Ω) can be identified with the dual space of C(Ω). The theory of operator semigroups has been so attractive to stochastic processes, because C0 -semigroups have an infinitesimal generator which allows the transition functions to be characterized as solutions of two different types of operator differential equations, known as forward and backward equations. It is one of the purposes of this paper to get rid of the restriction (•) and still preserve these characterizations of

Transition functions and semigroups

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transition functions. Other approaches can be found in [15, 17]. We use a more recent development in semigroup theory, integrated semigroups [1]. If we define Z t (1.4) [T (t)µ](D) = [S(r)µ](D)dr, 0

the operator family {T (t)} is a (once) integrated semigroup (Section 6). In turn, S can be characterized in terms of T by the relation T (t)S(r) = T (t + r) − T (r). S is uniquely determined by T provided T is non-denegerate, i.e., if µ 6= 0 then T (t)µ 6= 0 for at least one t > 0. S is called the integral semigroup associated with T (Section 6.1). Non-degeneracy holds under mild restrictions on K (Section 7) which follow from stochastic continuity (Section 3) if Ω is normal or σ-compact. A construction on BM(Ω) analogous to (1.4), which uses S˜ instead of S, typically leads to degenerate integrated semigroups and is less fruitful. A non-degenerate integrated semigroup is associated with a generator A, the generalization of the infinitesimal generator of a C0 -semigroup. In our case, the generator A is a Hille-Yosida operator in M(Ω). The transition function K is uniquely determined as the solution of the integral equation Z t (1.5) K(t, x, ·) = δx + A K(s, x, ·)ds, t ≥ 0, x ∈ Ω. 0

Here δx is the Dirac measure concentrated at x ∈ Ω. This equation can be formally rewritten as a differential equation which corresponds to the (Kolmogorov) forward equation (cf. [16, Sec.X.3 (3.5)], [15, Chap.4 (9.58)]). In turn, A is uniquely determined by the transition function as its resolvent can be expressed by the Laplace transform of K, Z Z ∞ −1 dte−λt K(t, x, ·), µ ∈ M(Ω). (1.6) (λ − A) µ = µ(dx) Ω

0

There is a subspace BM(Ω)◦ of BM(Ω) which is total for M(Ω) and ˜ to invariant under the semigroup S˜ such that the restrictions of S(t) ˜ BM(Ω)◦ form a C0 -semigroup S◦ . The infinitesimal generator of S˜◦ , ˜ A˜◦ , is dual R to A and has the following property: If f ∈ D(A◦ ) and v(t)(x) = Ω K(t, x, dy)f (y), then v is the unique solution of (1.7)

v 0 = A˜◦ v,

v(0) = f,

which corresponds to the (Kolmogorov) backward equation (cf. [16, Sec.X.3 (3.3), Sec.X.10], [24, Thm.7.11]). K is uniquely determined by this operator differential equation. For compact Ω, BM(Ω)◦ contains

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C(Ω) if the transition function K satisfies a sufficiently strong type of stochastic continuity (Section 5). While the riddance of (•) is an obvious goal for theoretical reasons, it is also important for the application of transition functions to deterministic models of structured populations, where this restriction is in the way of a satisfactory perturbation theory which would allow death and birth to be incorporated in addition to individual growth [9, 20, 22, 23]. 2. Transition functions and operator semigroups In the tradition of [21, Def.8.3.5], a (one-parameter) semigroup on a vector space X is a family of linear transformations S(t), t > 0, satisfying (2.1)

S(t + r) = S(t)S(r) ∀t, r > 0.

2.1. B0 -semigroups. All semigroups S we are going to consider here operate on a Banach space X and will satisfy the extra condition (2.2)

lim sup kS(t)xk < ∞ ∀x ∈ X. t&0

We call these semigroups B0 -semigroups. It follows from the uniform boundedness principle and from (2.1) that any B0 -semigroup is exponentially bounded, i.e., there exist M ≥ 1, ω ∈ R such that kS(t)k ≤ M eωt

∀t > 0.

2.2. C0 -semigroups. A semigroup S is called a C0 -semigroup if (2.3)

kS(t)x − xk → 0,

t & 0, x ∈ X.

It is then convenient to extend S(t) to [0, ∞) by (2.4)

S(0)x = x,

x ∈ X.

With this extension, (2.1) holds for all t, r ≥ 0, and S(t) is strongly continuous in t ≥ 0. For C0 -semigroups the infinitesimal generator A is defined by (2.5)

Ax = lim (1/t)(S(t)x − x), t&0

x ∈ D(A),

with D(A) consisting of all elements x ∈ X for which this limit exists. Any C0 -semigroup is a B0 -semigroup and is exponentially bounded. If S is a B0 -semigroup on a Banach space X, one introduces the space (2.6)

X◦ = {x ∈ X; kS(t)x − xk → 0, t & 0}.

X◦ is a closed subspace of X that is invariant under S(t) for all t ≥ 0. The restriction of S to X◦ , S◦ , is a C0 -semigroup on X◦ .


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˜ be measurable spaces with 2.3. Transition functions. Let Ω and Ω ˜ respective σ-algebras B and B. ˜ :Ω ˜ × B → R is called a Definition 2.1 ([3, 10.3.1]). A function K measure kernel if ˜ ˜ (a) K(x, ·) is a non-negative measure on B for all x ∈ Ω. ˜ D) is a B-measurable ˜ (b) K(·, function for all D ∈ B. ˜ is called bounded if sup ˜ K(x, ˜ A measure kernel K Ω) < ∞. x∈Ω Definition 2.2. A function K : R+ × Ω × B → R is called a transition function if (a) K(t, x, ·) is a non-negative measure on B for all t ≥ 0, x ∈ Ω. (b) K(t, ·, D) is a B-measurable function for all t ≥ 0, D ∈ B. (In other words, K(t, ·) is a measure kernel.) (c) K(0, x, D) = 1 if x ∈ D and K(0, x, D) = 0 if x ∈ Ω \ D. (d) There exist δ, c > 0 such that K(t, x, Ω) ≤ c for all t ∈ [0, δ], x ∈ Ω. A transition function K is called a Markov transition function if it satisfies the Chapman-Kolmogorov equations (1.3). A transition function is called a transition kernel if K(·, D) is BR+ ×B measurable where BR+ is the σ-algebra of Borel sets on R+ and BR+ ×B is the product σ-algebra, in other words if K is a measure kernel. Remark 2.1. Sometimes the term ‘transition function’ is used such that the Chapman-Kolmogorov equations are included [15]. We follow the use in [26, Sec.3.2] and [18, Sec.2.1] though it may not be clear whether they use ‘Markov’ to highlight the Chapman-Kolmogorov equations or the assumption that K(t, x, Ω) ≤ 1 (or = 1) which we do not make (cf. [3, 10.3.1]). We use ‘Markov’ in order to emphasize the connection of the Chapman-Kolmogorov equations to Markov processes [2, Sec.2.2] [15, Ch.4 (1.9)]. If K is a transition function, (1.1) and (1.2) define families of bounded linear operators on the space of measures on B of bounded variation, M(Ω), and the space of bounded measurable functions, BM(Ω), respectively. The Chapman-Kolmogorov equations are equivalent to the semigroup property of these operator families. By property (d), kS(t)k ≤ c, ˜ kS(t)k ≤ c for all t ∈ [0, δ] and both S and S˜ are B0 -semigroups and so exponentially bounded. Lemma 2.3. There exist ω ∈ R, M ≥ 1 such that kS(t)k ≤ M eωt , ˜ kS(t)k ≤ M eωt , K(t, x, Ω) ≤ M eωt for all t ≥ 0, x ∈ Ω.

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Through

Z hµ, f i =

f dµ,

f ∈ BM(Ω), µ ∈ M(Ω),

Ω

BM(Ω) can be identified with a subspace of M(Ω)∗ , the topological dual of M(Ω). Actually BM(Ω) is a sequentially weakly∗ closed subspace which norms M(Ω), i.e. sup{|hf, µi|; kf k ≤ 1} is an equivalent norm on M(Ω). We see that the semigroup of dual operators S ∗ (t) leaves BM(Ω) invariant and the restrictions of S ∗ (t) to BM(Ω) coin˜ cide with S(t). 2.4. A first shot at the forward and backward equations. Since S and S˜ are B0 -semigroups on M(Ω) and BM(Ω) respectively, we can use the construction (2.6) and obtain C0 -semigroups S◦ and S˜◦ on M(Ω)◦ and BM(Ω)◦ respectively with infinitesimal generators A◦ and A˜◦ . These operators are in duality, as hA◦ µ, f i = hµ, A˜◦ f i ∀µ ∈ D(A◦ ), f ∈ D(A˜◦ ). The equations (2.7)

u0 = A◦ u

and

v 0 = A˜◦ v

correspond to the (Kolmogorov) forward and backward equations. The backwards equation v 0 = A˜◦ v uniquely determines the transition function K if BM(Ω)◦ is a total subspace of M(Ω)∗ , i.e. it separates measures: if µ ∈ M(Ω) is not the 0 measure, then there exists some f ∈BM(Ω)◦ such that hµ, f i 6= 0. One of the tasks of this paper consists in finding practical conditions for this to be the case. The forward equation in the form of u0 = A◦ u is less useful because it seems difficult to come up with reasonable conditions which make M(Ω)◦ separate points in BM(Ω). One of the reasons that this is difficult lies in the fact that an important class of Markov transition functions is of the form K(t, x, dy) = κ(t, x, y)dy, with Ω being a measurable subset of Rn . Then S(t) maps M(Ω) into the closed subspace of measures which are Lebesgue absolutely continuous and which can be identified with L1 (Ω). This implies that M(Ω)◦ is contained in L1 (Ω). But L1 (Ω) does not separate the zero function from functions which are 0 Lebesgue almost everywhere. For the same class of Markov transition functions, the semigroup S˜ is degenerate, i.e. if f is 0 almost everywhere (but not ˜ everywhere), we still have that S(t)f is the zero-function in BM(Ω) for ˜ all t > 0. This makes S by itself not such a useful object of study and is one of the motivations to consider integrated and integral semigroups on M(Ω).


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2.5. Feller’s warning. Feller notes in his classic [16, X.8] that semigroups on M(Ω) seem to be a natural path to study Markov transition functions but warns that general semigroups on M(Ω) may be too large a class to work with as not all of the semigroups are linked to transition functions. The following holds, however. Proposition 2.4. There is a bijective correspondance between Markov transition functions and those B0 -semigroups on M(Ω) the duals of which leave BM(Ω) invariant. Proof. One direction of the correspondence is clear from the previous considerations. Now let S(·) be a B0 -semigroup on M(Ω) such that, for t ≥ 0, S ∗ (t) maps BM(Ω) into itself. We define K(t, x, D) = [S ∗ (t)χD ](x),

x ∈ Ω, D ∈ B.

Then K(t, x, D) is a measurable function of x ∈ Ω. Now K(t, x, D) = hδx , S ∗ (t)χD i = hS(t)δx , χD i = [S(t)δx ](D), which shows that K(t, x, ·) is a measure on B. Let µ ∈ M(Ω). Then Z ∗ [S(t)µ](D) = hS(t)µ, χD i = hµ, S (t)χD i = µ(dx)K(t, x, D). Ω

The other properties of a Markov transition function are easily checked for K. ¤ 3. Stochastic continuity of transition functions Little progress can be made in the interface of Markov transition functions and one-parameter semigroups unless the state space Ω is a topological Hausdorff space. Exceptions are transition functions associated with Markov jump processes [16, X.3] [15, 4.1] which induce C0 -semigroups on M(Ω) when Ω is just a measurable space [30]. Recall the operator semigroups S˜ on BM(Ω) and S on M(Ω), introduced in (1.2) and (1.1), associated with a Markov transition function K. Let Ω be a topological Hausdorff space. Cb (Ω) denotes the Banach space of real-valued bounded continuous functions on Ω with supremum norm. C0 (Ω) denotes the set of continuous real-valued functions that vanish at infinity. The latter means that for every ² > 0 there is a compact subset C of Ω such that |f (x)| < ² whenever x ∈ Ω \ C. When Ω is a topological space, the following two σ-algebras are typically considered: the σ-algebra of Borel sets generated by the open subsets in Ω and the σ-algebra of Baire sets which is the smallest σalgebra such that all continuous functions from Ω to R are measurable. In a topological space, we exclusively consider the σ-algebra of Baire

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sets which is denoted by B. The reason is that we need all finite measures on B to be regular in an appropriate sense; conditions which make every Borel measure regular seem to imply that the Borel and Baire sets coincide. Remark 3.1. Borel and Baire sets coincide if Ω is a metric space [3, 7.2.4], in particular if Ω is a locally compact space with countable base [3, 7.6.2, 7.6.3] or if, more generally, Ω is perfectly normal. Recall that a topological space is perfectly normal if it is normal and every open set is an Fσ -set, i.e. a countable union of closed sets [4, Sec.11 Exc.10]. In a normal space, the Baire-σ-algebra is generated by the open Fσ -sets [3, 7.2.3]. So, if every open set is an Fσ -set, the Baire and Borel σ-algebras coincide. 3.1. Locally compact state space. Let Ω be a locally compact space, i.e. it is a Hausdorff space and every x ∈ Ω is contained in an open set with compact closure. Definition 3.1. A measure kernel K is called weakly stochastically continuous, if for every open Kσ -set U ⊆ Ω with compact closure lim K(t, x, U ) = 1 whenever x ∈ U. t→0

˜ → R is upper semi-continuous at a Definition 3.2. A function f : Ω ˜ if the set {x ∈ Ω; ˜ f (x) < f (x0 ) + ²} point x0 in the topological space Ω is open for every ² > 0. f is called lower semi-continuous at x0 if −f is upper semi-continuous at x0 . Theorem 3.3. The following are equivalent for a locally compact space Ω and a Markov transition function K: (a) K is weakly stochastically continuous. (b) If U is an open Kσ set and x ∈ Ω, then K(·, x, U ) is lower semi-continuous at t = 0. If D is a compact Gδ set and x ∈ Ω, then K(·, x, D) is upper semi-continuous at t = 0. ˜ (c) For each x ∈ Ω, f ∈ C0 (Ω), [S(t)f ](x) is a right-continuous function of t ≥ 0. Proof. (a) =⇒ (b): The first part of (b) is an obvious consequence of (a). Let D be a compact Gδ -set. Let x ∈ Ω. There exists a function f ∈ C0 (Ω) such that f = 1 on D and f (x) = 1. Set U = {f > 1/2}. Then U is an open Kσ -set, x ∈ U , D ⊆ U , and U¯ ⊆ {f ≥ 1/2} is compact. By assumption, K(t, x, D) ≤ K(t, x, U ) → 1,

t→0+.

Now let x ∈ Ω \ D. Since x ∈ U , x is an element of the open S set U \ D. We claim that U \D is a Kσ set. Since U is a Kσ -set, U = ∞ n=1 Cn for a


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countable family of compact sets Cn . Since D is a Gδ -set, D = for a countable family of open sets Uk . Now n ∞ \ [ U \ D =U \ ( Uk ) = (U \ Uk ) k=1

=

∞ µ³ [ ∞ [ n=1

k=1

k=1

´ C n \ Uk

¶ =

∞ ∞ µ[ [ k=1

Tn k=1

Uk

¶ (Cn \ Uk ) .

n=1

So U \ D is the union of the countable family {Cn \ Uk ; k, n ∈ N} of compact sets Cn \ Uk . By assumption, since x ∈ U \ D, as t → 0+, K(D, t, x) = K(U, t, x) − K(U \ D, t, x) → 1 − 1 = 0 = K(D, 0, x). So K(D, t, x) is upper semi-continuous at t = 0. (b) =⇒ (c): (b) can be reformulated as follows: ˜ If U is an open Kσ -set, [S(t)χ U ](x) is upper semi-continuous at t = 0 ˜ for every x ∈ Ω. Further, if C is a compact Gδ set, [S(t)χ C ](x) is lower semi-continuous at t = 0 for every x ∈ Ω. The same proof as in ˜ Theorem A.2 shows that [S(t)f ](x) is right continuous at t = 0. Since ˜ S is a semigroup, Z ˜ ˜ ˜ ˜ [S(t + h)f ](x) = [S(t)S(h)f ](x) = [S(h)f ](y)K(t, x, dy). Ω

˜ Since [S(h)f ](y) → f (y) as h & 0, pointwise in x ∈ Ω, the dominated ˜ ˜ convergence theorem implies that [S(t+h)f ](x) → [S(t)f ](x) as h & 0, pointwise in x ∈ Ω. (c) =⇒ (a): Let x ∈ U and U open. Since Ω is locally compact, there exists a function f ∈ C0 (Ω) such that f (x) = 1, 0 ≤ f ≤ χU . Then Z K(t, x, U ) ≥

K(t, x, y)f (y)dy → f (x) = 1,

t → 0.

Ω

Since U has compact closure, there exist some f ∈ C0 (Ω) such that f (y) = 1 for all y ∈ U . So, for x ∈ U , ˜ K(t, x, U ) ≤ [S(t)f ](x) → f (x) = 1, t → 0. ¤ Ω is called σ-compact [12, XI.7] if it is locally compact and a Kσ set (countable at infinity [3, 7.4.4, 7.5]). Every σ-compact space is paracompact [12, XI.7] and every paracompact space is normal [12, VIII.2]. Corollary 3.4. Let Ω be σ-compact. Then, if the Markov transition function K is weakly stochastically continuous, it is a transition kernel.

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˜ Proof. Let f ∈ C0 (Ω). By Theorem 3.3, [S(t)f ](x) is right continuous R in t ≥ 0 and Baire measurable in x ∈ Ω. By Proposition B.1, f (y)K(t, x, dy) is jointly Baire-measurable in (t, x). Let U be an Ω open Kσ -set. Then there exists a sequence (fn ) in C0 (Ω) such that fn % R χU pointwise on Ω as n → ∞. So K(t, x, U ) = limn→∞ Ω fn (y)K(t, x, dy) is Baire-measurable in (t, x). It follows from our assumptions that the σ-algebra of Baire sets is generated by the open Kσ -sets [3, 7.1.3, 7.4.3, 7.4.5]. This implies that K(t, x, U ) is Baire measurable in (t, x) for all U ∈ B. ¤ A similar proof shows that C0 (Ω) separates point in M(Ω). Proposition 3.5. Let Ω be a σ-compact space. Then C0 (Ω) is a total subspace of M(Ω)∗ . As we mentioned before, we have a greater interest in the semigroup S than in the semigroup S˜ because the latter is degenerate for an important class of Markov transition functions. We therefore reformulate Theorem 3.3 in terms of S using Lebesgue’s theorem of dominated convergence. Corollary 3.6. The following are equivalent for a locally compact space Ω and a Markov transition function K: (a) K is weakly stochastically continuous. (b) For all f ∈ C0 (Ω) and µ ∈ M(Ω), hf, S(t)µi is a right-continuous function of t ≥ 0. 3.2. General topological Hausdorff spaces. Definition 3.7. A transition function K is called stochastically continuous if for every x ∈ Ω and every open Baire-set U ⊆ Ω, K(x, t, U ) → 1,

t → 0+,

whenever x ∈ U.

Theorem 3.8. (a) If K is stochastically continuous, then, for all f ∈ Cb (Ω) and µ ∈ M(Ω), hf, S(t)µi is a right continuous function of t ≥ 0. (b) If Ω is a completely regular space (e.g. a metric, normal, or locally compact space), the stochastic continuity of K is equivalent to the right continuity of hf, S(t)µi in t for all f ∈ Cb (Ω), µ ∈ M(Ω). Proof. (a) This part can be proved similarly as for Theorem 3.3 and Corollary 3.6. (b) Let x ∈ Ω be fixed but arbitrary. Let U 3 x be an open subset of Ω. Since Ω is completely regular, there exists a continuous function f : Ω → [0, 1] such that f (x) = 1 and f (y) = 0 for y ∈ Ω \ U . Then


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f ∈ Cb (Ω) and f ≤ χU . Assume that hf, S(t)µi is continuous in t ≥ 0 for µ = δx , the Dirac measure concentrated at x. Then Z K(t, x, U ) ≥ f (y)K(t, x, dy) → 1, t→0+. Ω

Since χΩ ∈ Cb (Ω),

Z

0 ≤ K(t, x, U ) ≤

χΩ (y)K(t, x, dy) → χΩ (x) = 1,

t→0+.

Ω

We combine the two statements and K(t, x, U ) → 1 as t → 0+.

¤

Corollary 3.9. Let Ω be a normal Hausdorff space. Then Cb (Ω) is a total subspace of M(Ω)∗ . If the Markov transition function K(t, x, B) is stochastically continuous, it is a transition kernel. Proof. Since Ω is normal, the σ-algebra of Baire sets is generated by the set of closed Gδ sets (and also by the set of open Fσ sets) [3, 7.2.3]. By Urysohn’s characterization of normality, Cb (Ω) separates disjoint closed sets. The proof proceeds now as the one of Corollary 3.4. ¤ 4. Which transition functions induce C0 -semigroups on C0 (Ω)? Often semigroup theory has been applied to Markov transition functions by assuming the Feller property, namely that the appropriate space of continuous functions is invariant under S˜ and a C0 -semigroup is induced [16, X.10], [19, 9.11], [26, 3.2]. In this section, we clarify the restrictions that the Feller property imposes on the transition function. Theorem 4.1. Let Ω be a locally compact Hausdorff space. Then a Markov transition function induces a C0 -semigroup on C0 (Ω) if and only if the following are satisfied. (i): For every compact Gδ -set C in Ω and every t > 0, K(t, ·, C) is upper semi-continuous on Ω. (ii): For every open Kσ -set U in Ω and every t > 0, K(t, ·, U ) is lower semi-continuous on Ω. (iii): For every compact subset C of Ω, every t > 0, and every ² > 0, there exists a compact subset C˜ of Ω such that K(t, x, C) < ² ˜ for all x ∈ Ω \ C. (iv): K is weakly stochastically continuous. A similar statement can be found in [31, Thm 2.1]. We are not able to prove the necessity-statement there (which has been left to the reader) unless it is interpreted in the sense above. We need the following abstract result [7, Prop.1.23][14, Ch.I, Thm.5.8].

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Theorem 4.2. Let {S(t); t ≥ 0} be a semigroup of bounded linear operators on a Banach space X. Then S(·) is a C0 -semigroup if (and only if ) it is weakly continuous at t = 0, i.e., hS(t)x, x∗ i → hx, x∗ i for t & 0, x ∈ X, x∗ ∈ X ∗ . Proof of Theorem 4.1. The necessity of (i), (ii), (iii) follows from Pro˜ maps C0 (Ω) into position A.2. (Sufficiency:) By Proposition A.2, S(t) ¦ ˜ itself for every t ≥ 0. Let S (t) be the restriction of S(t) to C0 (Ω). ¦ ˜ By Theorem 3.3, for f ∈ C0 (Ω) and x ∈ Ω, [S (t)f ](x) = [S(t)f ](x) is a continuous function of t at t = 0. It follows from the dominated convergence theorem that Z Z ¦ [S (t)f ](x)µ(dx) → f (x)µ(dx), t → 0, Ω

Ω

for every non-negative Borel measure µ on Ω and also for every signed Borel measure of finite variation. Since C0 (Ω)∗ can be identified with the Banach space of signed regular Borel measures of finite variation [4, Thm.38.7], S ¦ (t)f is weakly continuous in t at t = 0. By Theorem 4.2, S ¦ (·) is a C0 -semigroup. ¤ Analogously one can characterize the Markov transition functions which induce C0 -semigroups on Cb (Ω) for a normal space Ω by using Theorem 3.8 and Proposition A.3. ˇ Remark 4.1. By functional analytic magic (the Krein-Smulian theorem that the closed convex hull of a weakly compact set is weakly compact [13, V.6.Thm.3]), the weak stochastic continuity of K (together with the Feller property) implies the following locally uniform time-continuity statements: (i) Let C ⊂ U ⊂ Ω and C compact and U open. Then lim inf inf K(t, x, U ) ≥ 1. t→0

x∈C

(ii) Let C ⊂ Ω and C compact. Then lim sup inf K(t, x, C) ≤ 1. t→0

x∈C

If Ω is a locally compact metric space, C a compact set in Ω and K(t, x, Ω) ≤ 1 for all t ≥ 0, x ∈ Ω, then it follows [26, Thm. 3.1] that £ ¤ lim sup sup 1 − K(t, x, B² (x)) = 0, t&0

x∈C

for all ² > 0 where B² (x) is the open ball with center x and radius ². As far as sufficient conditions are concerned, Theorem 4.1 shows that this assumption in [26, Thm. 3.1] can be considerably relaxed (and the separability of Ω be dropped).


13

Proof. (i) Choose f ∈ C0 (Ω) with f = 1 on C, f = 0 on U , and 0 ≤ f (y) ≤ 1 for all y ∈ Ω. Since S˜ induces a C0 -semigroup on C0 (Ω), ¯Z ¯ ¯ ¯ sup¯ K(t, x, dy)f (y) − f (x)¯ → 0, t → 0. x∈Ω

Ω

Let ² > 0. Then there exists some δ > 0 such that Z f (x) + ² > K(t, x, dy)f (y) > f (x) − ² ∀t ∈ [0, δ]. Ω

For x ∈ C,

Z

1 − ² = f (x) − ²

f (y)K(t, x, dy) = K(t, x, C). C

¤ An obvious example of a Markov transition function which does not have the Feller property is K(t, x, D) = e−tγ(x) δx (D) where γ is a nonnegative Baire measurable function which is not continuous. Then the ˜ associated semigroup on BM(Ω), [S(t)f ](x) = f (x)e−tγ(x) , does not map continuous functions f to continuous functions. In the following example, with Ω = R+ , the semigroup S˜ preserves continuity, but not the property of vanishing at infinity. We interpret exponential growth, N (t) = N0 e(β−µ)t , in an age-structured population model: β and µ are the per capita birth and mortality rates, N (t) the population size at time t and N0 the initial stratify R ∞ population size. RWe ∞ the population along age: N (t) = 0 u(t, a)da, N0 = 0 u0 (a)da. Then u satisfies the McKendrick equation (4.1)

ut + ua = −µu,

u(t, 0) = βN (t),

u(0, a) = u0 (a),

where ut and ua are the partial derivatives with respect to t and a. One readily checks that  u0 (a −Zt)e−µt ; t < a,  ∞ u(t, a) = u0 (s)ds; t > a,  βeβ(t−a) e−µt 0

is a solution of (4.1) for t 6= a if u0 is differentiable. Otherwise it is a solution in an appropriately generalized sense. In this example, the semigroup S leaves L1 (R+ ) invariant (identified with the subspace of measures which are absolutely continuous relative to the Lebesgue

14


˜ measure) and S(t)u0 = u(t, ·). We use the formal duality hS(t)f, u0 i = hf, S(t)u0 i to find Z t ³ ´ −µt β(t−s) ˜ (4.2) [S(t)f ](a) = e f (a + t) + f (s)e ds , 0

˜ maps continuous functions to continfor f ∈ BM(R+ ). For t > 0, S(t) ˜ uous functions, but lim [S(t)f ](a) > 0 if f ∈ C0 (R+ ) is positive. To a→∞

make this example rigorous, we observe that (4.2) defines a semigroup S˜ on BM(R+ ) indeed and that the associated transition function is Z ³ ´ −µt K(t, a, D) = e δt+a (D) + eβ(t−s) ds . [0,t]∩D

Similar problems arise in body-size structured population models which involve per capita birth rates β(x) that depend on body size x. Constructing the associated C0 -semigroup [10] (strongly continuous evolutionary system [11]) on C0 (R+ ) by perturbation requires β(x) → 0 as x → ∞, i.e. the birth rate must tend to 0 for large body sizes. This assumption which may be unrealistic in certain applications can be dropped if the Feller property does not need to be satisfied [20, 23]. 5. More continuity results and a backward integral equation Motivated by the characterization of the Feller property in the previous section and its failure in the preceding examples, we want to work without it and the restrictions it imposes on Markov transition kernels. Recall the space BM(Ω)◦ of those bounded measurable functions f on ˜ Ω for which S(t)f is a continuous function of t; the Markov transition function K induces the C0 -semigroup S˜◦ on BM(Ω)◦ (Section 2.4 and (2.6)). In the following we derive conditions for C0 (Ω) ⊆BM(Ω)◦ if Ω is locally compact. This is of interest as the domain of infinitesimal generator A˜◦ of S˜◦ involved in the backward equation (1.7) (see also Section 2.4) is hard to characterize and one can alternatively consider the integral version Z t (5.1) v(t) = f + A˜◦ v(r)dr, f ∈ BM(Ω)◦ , 0

R which is uniquely solved by v(t)(x) = Ω f (y)K(t, x, dy). Throughout this section, Ω is a locally compact Hausdorff space and K a Markov transition function. Further we assume that


15

(¦) for every ² > 0, x ∈ Ω, there exist an open neighborhood U of x and δ > 0 such that K(t, z, Ω) ≤ 1 + ²

∀z ∈ U, t ∈ [0, δ].

In many stochastic applications, condition (¦) is trivially satisfied as K(t, x, Ω) ≤ 1. The perturbation theory in [23] leads to Markov transition kernels which only satisfy (¦), but not this more restrictive inequality. In the next results, we investigate a stronger stochastic continuity concept than the ones used in Section 3. Theorem 5.1. C0 (Ω) ⊆ BM(Ω)◦ if and only if the following two statements hold: (i) If U ⊂ Ω is an open Baire set, x ∈ U and ² > 0, then there exist δ > 0 and an open set U˜ 3 x such that K(t, z, U ) ≥ 1 − ² whenever z ∈ U˜ , 0 ≤ t < δ. (ii) If C ⊆ Ω is a compact Baire set and ² > 0, then there exist a compact set C˜ ⊆ Ω and δ > 0 such that ˜ 0 ≤ t < δ. K(t, x, C) ≤ ² ∀x ∈ Ω \ C, Corollary 5.2. Make the additional assumptions (i), and (ii) of Theorem 5.1. Then C0 (Ω) ⊆ BM(Ω)0 and, R t for every f ∈ C0 (Ω), the back˜ ward integral equation R v(t) = f + A0 0 v(r)dr has the unique continuous solution v(t) = Ω K(t, ·, dy)f (y). If Ω is σ-compact, the transition function K is uniquely determined by this fact. The solution v does not necessarily take values in C0 (Ω). The first part of the corollary follows directly from the previous theorem, the properties of BM(Ω)0 , and standard semigroup theory. The uniqueness of K follows from Proposition 3.5. Proof of Theorem 5.1. ‘⇒’: Let U ⊆ Ω be open, x ∈ U and ² > 0. Since Ω is locally compact, there exist an open set U˜ and a compact set C such that x ∈ U˜ ⊆ C ⊆ U. Again, since Ω is locally compact, there exist some f ∈ C0 (Ω) such that χC ≤ f ≤ χU . Then ˜ K(t, z, U ) ≥ [S(t)f ](z) → f (z), t & 0, uniformly in z ∈ Ω. Since f (z) ≥ 1 for all z ∈ U˜ , (i) follows. Let C ⊆ Ω be compact. Since Ω is locally compact, there exist an open set U and a compact set C˜ such that C ⊆ U ⊆ C˜ ⊆ Ω. Again, since Ω is locally compact, there exist f ∈ C0 (Ω) such that χC ≤ f ≤ χU . Then ˜ K(t, x, C) ≤ [S(t)f ](x) → f (x), t & 0,

16


˜ (ii) follows. uniformly in x ∈ Ω. Since f (x) = 0 for all x ∈ C, ‘⇐’: By the semigroup property, it is sufficient to show that, for ˜ f ∈ C0 (Ω), S(t)f is continuous at t = 0. Without loss of generality, ˜ we can assume that f is a non-negative function. Suppose that S(t)f is not continuous at t = 0. Then there exist η > 0, a sequence tn & 0 and a sequence (xn ) in Ω such that ¯ ¯ ¯ ¯ ˜ ∀n ∈ N. (5.2) ¯[S(tn )f ](xn ) − f (xn )¯ > 2η We can assume that η ≤ 1. Claim: There exists a compact set C ⊆ Ω such that xn ∈ C for all n ∈ N. Suppose that the claim does not hold. By Definition 2.2 (d), there exists some δ0 > 0, c > 0 such that K(t, x, Ω) ≤ c

∀t ∈ [0, δ0 ], x ∈ Ω.

There exists some n0 ∈ N such that tn ≤ δ0 for all n ≥ n0 . Since f ∈ C0 (Ω), there exists some compact set C1 ⊆ Ω such that η (5.3) |f (x)| ≤ ∀x ∈ Ω \ C1 . 8(1 + c) For all x ∈ Ω, t ≥ 0,

Z

Z

˜ |S(t)(x)| ≤ (5.4)

|f (y)|K(t, x, dy) + C1

|f (y)|K(t, x, dy) Ω\C1

η ≤kf kK(t, x, C1 ) + K(t, x, Ω). 8 By (ii), there exists a compact set C2 ⊆ Ω and some δ1 > 0 such that η (5.5) K(t, x, C1 ) ≤ x ∈ Ω \ C2 , t ∈ [0, δ1 ]. 8kf k + 1

Set δ = min{δ1 , δ0 }, C = C1 ∪ C2 ∪ {x1 , . . . xn0 }. Then C is compact. Since (xn ) is not contained in any compact set, there exists some n ∈ N, n ≥ n0 , such that xn ∈ Ω \ C. By (5.3), (5.4), and (5.5), for such an xn we have ¯ ¯ ¯ ¯ ¯ ¯ ˜ n )f ](xn ) − f (xn )¯ ≤ ¯[S(t ˜ n )f ](xn )¯+¯f (xn )¯ ≤ η, ¯[S(t a contradiction. This proves the claim. ˜ n )f ] After choosing subsequences, the real sequences (f (xn )) and ([S(t T∞ (xn )) converge. It follows from the claim that m=1 {xn ; n ≥ m} 3 x for some x ∈ Ω. Since f is continuous and f (xn ) has a limit, f (xn ) → f (x) as n → ∞. By (5.2), there exists some mη ∈ N such that ¯ ¯ ˜ n )f ](xn ) − f (x)¯ ≥ η ¯[S(t (5.6) ∀n ≥ mη .


17

In order to derive a contradiction, we let U = {y ∈ Ω; |f (y) − f (x)| < η/8}. Then U is an open Baire set, x ∈ U , f (x) ≥ 0, and Z ¯ ¯ ˜ n )](xn )−f (x)¯ ≤ ¯[S(t |f (y) − f (x)|K(tn , xn , dy) U Z £ ¤ + |f (y)|K(tn , xn , dy) + |f (x)| K(tn , xn , Ω) − 1 + , Ω\U

where [r]+ = max{r, 0} is the positive part of a real number r. By the definition of U , ¯ ¯ £ ¤ ˜ n )](xn ) − f (x)¯ ≤ η K(tn , xn , U ) + kf k K(tn , xn , Ω) − 1 ¯[S(t + 8 + kf k(K(tn , xn , Ω) − K(tn , xn , U )). Since every open set U˜ ⊆ Ω with x ∈ U˜ has non-empty intersection with all sets {xn ; n > m}, m ∈ N, By (¦) and (i), there exists some n > mη such that η η K(tn , xn , Ω) < 1 + ≤ 2, K(tn , xn , U ) > 1 − . 8kf k + 1 8kf k + 1 For such an n > mη , we have ¯ ¯ ˜ n )f ](xn ) − f (x)¯ < 3η , ¯[S(t 4 a contradiction to (5.6). ¤ For a locally compact metric space, K is called locally uniformly stochastically continuous [26, 3.2] if for each δ > 0 and each compact subset C of Ω, K(t, x, Bδ (x)) → 1,

t → 0+, uniformly in x ∈ C.

Theorem 5.3. Consider the statements (i) and (ii) in Theorem 5.1 and the following statements: (iii) K is locally uniformly stochastically continuous. (iv) C0 (Ω) ⊆BM(Ω)◦ . Then we have the following equivalences, [(i) ∧ (ii)] ⇐⇒ [(ii) ∧ (iii)] ⇐⇒ (iv). Proof. The equivalence [(i) ∧ (ii)] ⇐⇒ (iv) has been established in Theorem 5.1. (iii) =⇒ (i): Let U be an open set and x ∈ U . Then there exists some η > 0 such that Bη (x) ⊆ U . Since Ω is locally compact, there exists an open set U˜ such that x ∈ U˜ and the closure of U˜ is compact.

18


So K(t, x, Bη (x)) → 1 as t → 0+ uniformly on U˜ . Let ² > 0. Then there exists some δ > 0 such that K(t, x, U ) ≥ K(t, x, Bη (x)) > 1 − ²

whenever t ∈ [0, δ], x ∈ U˜ .

(iv) =⇒ (iii) follows from the proof of [26, Thm.3.1]. ¤ The equivalence (ii) ∧ (iii) ⇐⇒ (iv) is proved directly in [26, Thm.3.1]. There it is assumed that K has the Feller property, i.e. S˜ leaves C0 (Ω) invariant, but this property is not really used. In fact if it is satisfied, weak stochastic continuity instead of (iii) (or (i)) is sufficient in combination with (ii) to make the restriction of S˜ to C0 (Ω) a C0 semigroup (see Theorem 4.1). For further continuity results we refer to [22]. 6. Integrated semigroups Our quest for a forward equation and a backward equation each of which uniquely characterizes the Markov transition function leads us to (once) integrated semigroups, T , strongly continuous operator families which satisfy Z t+r Z t Z r T (t)T (r) = T (s)ds − T (s)ds − T (s)ds, t, r ≥ 0, (6.1) 0 0 0 T (0) = 0. Rt These relations can be motivated by formally defining T (t) = 0 S(s)ds, with a semigroup S. For an authoritative survey and bibliographic notes concerning integrated semigroups we refer to [1, Chap.3]. One is mainly interested in non-degenerate integrated semigroups, i.e., T (t)x = 0 for all t > 0 occurs only for x = 0. The generator A of a non-degenerate integrated semigroup is given as follows [27]: if x, y ∈ X, Z t (6.2) x ∈ D(A), y = Ax ⇐⇒ T (t)x−tx = T (s)yds ∀t ≥ 0. 0

Notice that this definition makes sense and defines a closed operator A, even if T is not an integrated semigroup. Actually one has the following result: Theorem 6.1. Let T (t), t ≥ 0, be a non-degenerate strongly continuous family of bounded linear operators on X and let the closed linear


19

operator A Rbe defined by (6.2). Then T is an integrated semigroup if t and only if 0 T (s)ds ∈ D(A) for all t ≥ 0 and Z t A T (s)xds = T (t)x − tx ∀t ≥ 0. 0

Proof. The “only if” part follows from [27, Lemma 3.4]. The “if” part follows from the proof of [27, Thm.6.2]. ¤ If T (t) is exponentially bounded, i.e., there exist M, ω > 0 such that kT (t)k ≤ M eωt

∀t ≥ 0,

one has the following useful relation between the Laplace transform of the integrated semigroup and the resolvent of the generator. It follows by combining [1, Prop.3.2.4] and [27, Prop.3.10]. Theorem 6.2. Let T (t), t ≥ 0, be a strongly continuous exponentially bounded family of bounded linear operators on X and A : D(A) → X be a linear operator in X. Then T is a non-degenerate integrated semigroup and A its generator if and only if there exists some ω > 0 such that any λ > ω is contained in the resolvent set of A and the resolvent of A can be expressed in terms of Laplace transforms of T , Z ∞ −1 ˆ e−λt T (t)xdt. (6.3) (λ − A) = λT (λ) := λ 0

Actually formula (6.3) can be used to define the generator A in the case of exponentially bounded integrated semigroups [1, Def.3.2.1]. A particularly interesting family of (once) integrated semigroups are those that are locally Lipschitz continuous (l.L.c.) [1, Sec.3.5]. Theorem 6.3. The following statements (i), (ii), and (iii) are equivalent for a closed linear operator A in a Banach space X: (i) A is the generator of a non-degenerate integrated semigroup T that is l.L.c.: for any b > 0, there exists some Λ > 0 such that kT (t) − T (r)k ≤ Λ|t − r|,

0 ≤ r, t ≤ b.

(ii) A is the generator of a non-degenerate integrated semigroup T and there exist constants M ≥ 1, ω ∈ R such that Z t kT (t) − T (r)k ≤ M eωs ds, 0 ≤ r ≤ t < ∞. r

(iii) A is a Hille-Yosida operator, i.e., there exist M ≥ 1, ω ∈ R such that (ω, ∞) is contained in the resolvent set of A and k(λ − A)−n k ≤ M (λ − ω)−n ,

λ > ω, n = 1, 2, . . .

20


The constants M, ω in (ii), (iii) can be chosen to be identical. • Moreover, if one (and then all) of (i), (ii), (iii) holds, D(A) coincides with those x ∈ X for which T (t)x is continuously differentiable. The derivatives S◦ (t) = T 0 (t)x, t ≥ 0, x ∈ D(A), provide bounded linear operators S◦ (t) from X◦ = D(A) into itself forming a C0 -semigroup on X◦ which is generated by the part of A in X◦ , A◦ . Finally T (t) maps X into X◦ and (2.10)

T 0 (r)T (t) = T (t + r) − T (r),

r, t ≥ 0.

Proof: The statements follow from combining the results in [1, Chap.3]. Remark 6.1. X◦ = D(A) can be characterized in various ways: (6.4)

X◦ ={x ∈ X; kλ(λ − A)−1 x − xk → 0,

λ → ∞}

={x ∈ X; k(1/h)T (h)x − xk → 0, h & 0}.

We define the closed subspace X ¯ (pronounced “X sun”) of the dual space of X, X ∗ , (6.5)

X ¯ = {x∗ ∈ X ∗ ; kλ(λ − A)−1∗ x∗ − x∗ k → 0, λ → ∞}.

If we want to emphasize the dependence of X ¯ on the generator A, we write XA¯ . The resolvent identity implies that, for λ ∈ ρ(A), (λ − A)−1∗ maps X ∗ into X ¯ and actually X ¯ = (λ − A)−1∗ X ∗ . Notice that X ¯ separates points in X and norms X◦ : kx◦ k ≤ M sup{|hx◦ , x¯ i|; x¯ ∈ X ¯ , kx¯ k ≤ 1}. Vice versa, X◦ norms X ¯ . The restriction of (λ − A)−1∗ to X ¯ forms a family of pseudoresolvents that is actually the resolvent of a closed linear operator A¯ in X ¯ . It is easy to show that A¯ is densely defined in X ¯ and, of course, a Hille-Yosida operator, and thus the infinitesimal generator of a C0 -semigroup S ¯ on X ¯ . We have the following relations: X ¯ ={x∗ ∈ X ∗ ; k(1/h)T ∗ (h)x∗ − x∗ k → 0, h & 0} Z t ∗ ¯ (6.6) T (t)x = S ¯ (r)x¯ dr, t ≥ 0, x¯ ∈ X ¯ , 0

hS◦ (t)x◦ , x¯ i =hx◦ , S ¯ (t)x¯ i,

t ≥ 0, x◦ ∈ X◦ , x¯ ∈ X ¯ .

˜ be a total subspace of X ∗ , i.e. X ˜ separates Proposition 6.4. Let X ∗ ˜ such that points in X: if x ∈ X, x 6= 0, then there exists some x ∈ X ∗ ∗ ˜ is invariant under T (·), or equivalently hx, x i 6= 0. Assume that X −1∗ under (λ − A) .


21

˜ ∩ X ¯ is a total subspace of X ∗ . Further the C0 Then X ® := X semigroup S ¯ leaves X ® invariant and its restrictions form a C0 semigroup S ® on X ® . The generator of S ® is the part of A¯ in X ® , i.e. the restriction of A¯ to D(A® ) = {x¯ ∈ D(A¯ ) ∩ X ® ; A¯ x¯ ∈ X ® }. A acts like the dual operator of A® : iff x, y ∈ X, then (6.7) x ∈ D(A), Ax = y ⇐⇒ hx, A® x¯ i = hy, x¯ i ∀x¯ ∈ D(A® ). ˜ Proof. Let x ∈ X and hx, x∗ i = 0 for all x∗ ∈ X ® . Let y ∗ ∈ X. −1∗ −1∗ ∗ ® ˜ Since X is invariant under (λ − A) , (λ − A) y ∈ X . So 0 = ˜ hx, (λ − A)−1∗ y ∗ i = h(λ − A)−1 x, y ∗ i. Since this holds for all y ∈ X ˜ is a total subspace of X ∗ , (λ − A)−1 x = 0 and x = 0. Since and X (λ−A¯ )−1 is the restriction of (λ−A)−1∗ to X ¯ , it leaves X ® invariant and so does the C0 -semigroup S ¯ which is generated by A¯ . (6.7)‘⇒’ follows from the construction of A® . Let x, y satisfy the right hand side of (6.7). Let y ¯ ∈ X ® and set x¯ = (λ − A)−1∗ y ¯ = (λ − A® )−1 y ¯ for some sufficiently large λ > 0. Then x¯ ∈ D(A® ) and hy, (λ − A)−1∗ y ¯ i = hy, x¯ i = hx, A¯ x¯ i =hx, A¯ (λ − A¯ )−1 y ¯ i = hx, −y ¯ + λ(λ − A¯ )−1 y ¯ i. By duality, h(λ − A)−1 y, y ¯ i = h−x + λ(λ − A)−1 x, y ¯ i. Since y ¯ ∈ X ® has been arbitrary and X ® is a total subspace of X ∗ , (λ − A)−1 y = −x + λ(λ − A)−1 x. Thus x ∈ D(A) and Ax = y. ¤ 6.1. Integral semigroups. We start with the following observation concerning two operator families on a Banach space X [29, 2.4]. Lemma 6.5. Let S(t), T (t), t ≥ 0, be two families of bounded linear operators on X, T non-degenerate, T (0) = 0, such that (6.8)

T (r)S(t) = T (r + t) − T (t),

t, r ≥ 0.

Then S is uniquely determined by T and is a semigroup satisfying S(0)x = x for all x ∈ X. If T (t) is strongly continuous, then S is non-degenerate. Proof. S is uniquely determined by T because T is non-degenerate. By (6.8), T (r)S(t)S(u) = (T (r + t) − T (t))S(u) =T (r + t + u) − T (u) − (T (t + u) − T (u)) = T (r + t + u) − T (t + u) =T (r)S(t + u),

∀r, t, u ≥ 0.

22


As T is non-degenerate, S(t)S(u) = S(t + u). Similarly we conclude from T (r)S(0) = T (r) − T (0) = T (r) that S(0) = I. Now assume that T (t), t ≥ 0, is strongly continuous, x ∈ X, and S(t)x = 0 for all t > 0. By (6.8), T (r)x is constant on every interval [t, ∞), t > 0, and hence constant on (0, ∞). By continuity and T (0) = 0 we have that T (r)x = 0 for all r ≥ 0. Since T is non-degenerate, x = 0. ¤ Definition 6.6. Let S(t), t ≥ 0, be a family of bounded linear operators on X. S is called an integral semigroup if there exists a l.L.c. integrated semigroup T (t), t ≥ 0, such that T (r)S(t) = T (r + t) − T (t),

t, r ≥ 0.

S is called the integral semigroup associated with T . If A is the generator of T , S is called the integral semigroup generated by A. A closed linear operator A is called the generator of an integral semigroup if A generates a l.L.c. integrated semigroup T and there exists an integral semigroup S associated with T . The integral semigroup S is uniquely determined by the generator A because A uniquely determines T , and we will see later (Proposition 3.6) that S uniquely determines A and, equivalently, T . The definition of an integral semigroup also makes sense if the associated integrated semigroup is not l.L.c. The following theorem also holds in this more general case. Theorem 6.7. Let A be the generator of an integrated semigroup T . Then the following statements are equivalent: (i) A generates an integral semigroup. (ii) T (t) maps X into D(A) for all t ≥ 0. If one and then both statements hold we have the following relations for the integral semigroup S generated by A: S(t)x = x + AT (t)x,

x ∈ X, t ≥ 0.

Proof. (ii) ⇒ (i): Set S(t)x = x + AT (t)x. Since T (r) and A commute [27, L.3.4], T (r)S(t)x = T (r)x + AT (r)T (t)x. By (6.1) and Theorem 6.1, Z r Z t ³Z t+r ´ T (r)S(t)x =T (r)x + A T (u)xdu − T (u)xdu − T (u)xdu 0

0

0

= T (t + r) − T (t)x. Hence (6.8) holds and S is the integral semigroup generated by A.


23

(i) ⇒ (ii): We have to show that T (t) maps into D(A). We use the definition in (6.2). By (6.8) and (6.1), d T (r)(S(t)x − x) = T (t + r)x − T (r)x − T (t)x = T (r)T (t)x − T (t)x. dr After integration, by (6.2), T (t)x ∈ D(A) and AT (t)x = S(t)x − x. ¤ We now restrict our discussion to the case that the integral semigroup is associated with a l.L.c. integrated semigroup. We recall that X◦ = D(A) coincides with the space of those x ∈ X such that T (t)x is continuously differentiable in t ≥ 0 and that S◦ (t) = T 0 (t), t ≥ 0, form a C0 -semigroup on X◦ which is generated by the part of A in X◦ , A◦ . Lemma 6.8. Let x ∈ X. Then x ∈ X◦ if and only if S(t)x is continuous in t ≥ 0. Further S(t) extends S◦ (t) from X◦ to X and S◦ (t)T (r) = T (r)S(t) for all r, t ≥ 0 and S◦ (t)(λ − A)−1 = (λ − A)−1 S(t). Proof. Let x ∈ X and S(t)x be continuous in t ≥ 0. Then, by the integrated semigroup property, Z t Z t T (r) S(u)xdu = (T (r + u) − T (u))xdu = T (r)T (t)x. 0

0

Rt

As T is non-degenerate, 0 S(u)xdu = T (t)x. Hence T (t)x is continuously differentiable and dtd T (t)x = S(t)x. On the other hand, if T (t)x is continuously differentiable, then by the integrated semigroup property, d T (r) T (t)x = T (r + t)x − T (t)x = T (r)S(t)x. dt As T is non-degenerate, dtd T (t)x = S(t)x and S(t)x is continuous in t. The remaining statements follow from Theorem 6.3 and (6.3). ¤ Corollary 6.9. Let A be the generator of an integral semigroup S. Then S ∗ (t) extends S ¯ (t) from X ¯ to X ∗ . Proof. By Theorem 6.7, S is given by S(t)x = x + AT (t)x. Let x¯ ∈ D(A¯ ). Then hx, S ∗ (t)x¯ i = hS(t)x, x¯ i = hx + AT (t)x, x¯ i D Z t E ¯ ∗ ¯ ¯ ¯ =hx, x i + hx, T (t)A x i = hx, x i + x, S ¯ (r)A¯ xdr, x¯ 0

=hx, S ¯ (t)x¯ i. ¤ The example in [5, Sec.4] shows the following: If S ∗ (t)x∗ is continuous in t ≥ 0, then x∗ is not necessarily an element in X ¯ . However, the following holds:

24


Proposition 6.10. a) Let x∗ , y ∗ ∈ X ∗ . Then x∗ ∈ D(A¯ ) and A¯ x∗ = y ∗ if and only if (1/h)(S ∗ (h)x∗ − x∗ ) → y ∗ as h → 0. b) An integral semigroup uniquely determines its generator. Proof. (a) ‘⇒’ is obvious since S ∗ extends S ¯ . Now let y ∗ = limh&0 (1/h) (S ∗ (h)x∗ − x∗ ). By Corollary 6.9, (λ − A)−1∗ y = lim (1/h)(S ∗ (h) − I)(λ − A)−1∗ x∗ h&0

= lim (1/h)(S ¯ (h) − I)(λ − A)−1∗ x∗ h&0

=A¯ (λ − A)−1∗ x∗ = −x∗ + λ(λ − A)−1∗ x∗ . This implies that x∗ ∈ X ¯ and y ∗ = limh&0 (1/h)(S ¯ (h)x∗ − x∗ ), hence x∗ ∈ D(A¯ ), y ∗ = A¯ x∗ . (b) By (a), A¯ is uniquely determined by S ∗ and thus by S. Since A¯ uniquely determines A by (6.7), A is uniquely determined by S. ¤ ˜ be a subspace of X ∗ which separates points in Proposition 6.11. Let X ˜ Let {S(t); t ≥ 0} X and Y a subset of X which separates points in X. and {T (t); t ≥ 0} be families of bounded linear operators such that S ∗ (t) ˜ into itself for all t ≥ 0. Assume that T (t) is strongly and T ∗ (t) map X continuous in t ≥ 0 and hS(t)x, x∗ i is Borel measurable in t ≥ 0 for all ˜ and x ∈ X and x∗ ∈ X Z t ∗ ˜ hS(r)x, x∗ idr ∀x ∈ X, x∗ ∈ X. (6.9) hT (t)x, x i = 0

Finally let A be a Hille-Yosida operator in X such that (λ−A)−1∗ maps ˜ into itself for all λ > ω and S(t)x = x+AT (t)x for all t ≥ 0, x ∈ Y . X Then T is an integrated semigroup and A its generator and S the associated integral semigroup, further Z ∞ −1 ∗ ˜ h(λ − A) x, x i = e−λt hS(t)x, x∗ idt, λ > ω, x ∈ X, x∗ ∈ X. 0

Proof. Recall Proposition 6.4. Let x¯ ∈ D(A® ). By (6.7), hx, S ∗ (t)x¯ i = hx, x¯ i + hx, T ∗ (t)A® x¯ i for all x ∈ Y . Since S ∗ (t)x¯ ∈ X ® and T ∗ (t)A® x¯ ∈ X ® and Y ˜ ⊇ X ®, separates point in X S ∗ (t)x¯ = x¯ + T ∗ (t)A® x¯ . For all x ∈ X, by duality, (6.10)

hS(t)x, x¯ i = hx, x¯ i + hT (t)x, A® x¯ i.


25

By Proposition 6.4, T (t)x ∈ D(A) and (6.11)

S(t)x = x + AT (t)x.

We integrate (6.10) in time and use (6.9), DZ t E ¯ ¯ hT (t)x, x i = thx, x i + T (r)xdr, A® x¯ . 0

Again by Proposition 6.4, (6.7), Z

t

T (t)x = tx + A

T (r)xdr. 0

Since A is a Hille-Yosida operator, it generates an integrated semi˜ group T˜ (Theorem R t 6.3) and u(t) = T (t)x is the unique solution of u(t) = tx + A 0 u(r)dr [27, Thm.6.1]. This implies T = T˜ and T is an integrated semigroup and A its generator. By (6.11) and Theorem 6.7, S is the integral semigroup associated with T . The relation between the resolvent of A and the Laplace transform of S follows from Theorem 6.2 and the assumed relation between T and S. ¤ 7. Markov transition functions and integral semigroups For a Markov transition kernel K we can define the operator families Z t ³Z ´ (7.1) [T (t)µ](D) = µ(dx)K(s, x, D) ds, µ ∈ M(Ω), D ∈ B. 0

Ω

˜ Since K is jointly measurable, [S(t)f ](x) in (1.2) is jointly measurable in (t, x) and we can change the order of integration ad libitum. This implies that T (·) is an integrated semigroup and S the associated integral semigroup, Z t hf, S(r)µidr ∀f ∈ BM(Ω), µ ∈ M(Ω). (7.2) hf, T (t)µi = 0

Further the dual integrated semigroup T ∗ leaves BM(Ω) invariant and Z t ³Z Z t ´ ∗ ˜ [T (t)f ](x) = K(t, x, dy)f (y) ds = [S(s)f ](x)ds, (7.3) 0 Ω 0 t ≥ 0, x ∈ Ω, f ∈ BM(Ω). See [22] for details. We mention that there are one-parameter semigroups S on M(Ω) (not induced by a Markov transition kernel) for which (1.4) does not lead to an integrated semigroup T [5, Sec.1].

26


7.1. Non-degeneracy. A transition function K is called non-degenerate if a measureR µ ∈ M(Ω) is necessarily the zero measure whenever, for any D ∈ B, Ω K(t, x, D)µ(dx) = 0 for almost all t > 0. Obviously a Markov transition kernel K is non-degenerate if and only if the associated integrated semigroup is non-degenerate. We derive topological conditions for K to be non-degenerate. Theorem 7.1. Let Ω be a topological space and B the Baire σ-algebra. Then a Markov transition function K is a non-degenerate Markov transition kernel if one of the following conditions hold: (a) Ω is a normal space and K is stochastically continuous (b) Ω is σ-compact and K is weakly stochastically continuous. Proof. (a) The joint measurability has already been proved in Corollary 3.9. Let µ ∈ M(B) and T (t)µ = 0 for all t ≥ 0. Since hf, S(t)µi is right-continuous in t ≥ 0, it follows from (7.2) that hf, µi = 0 for all f ∈ Cb (Ω) and µ = 0 because Cb (Ω) is a total subspace of M(Ω) by Corollary 3.9. (b) The proof is similar to the one for (a) except that we use Proposition 3.4 and Proposition 3.5. ¤ 7.2. Forward and backward equations. Let us now assume that Ω is a measurable space with σ-algebra B and K is a non-degenerate Markov transition kernel. Then the generator, A, of the associated non-degenerate integrated semigroup, T , is defined and a Hille-Yosida operator, and the resolvents of A and the Laplace transform of T are related by Theorem 6.2. By (7.1), this yields the following relation. Theorem 7.2. Let K be a non-degenerate Markov transition kernel. Then there exists a Hille-Yosida operator A in M(Ω) such that Z Z ∞ −1 [(λ − A) µ](D) = µ(dx) dte−λt K(t, x, D) Ω

0

for all λ > ω, D ∈ B, µ ∈ M(Ω). Through this formula, K and A determine each other in a unique way. Obviously A is uniquely determined by this equation. In turn, the integrated semigroup T is uniquely determined by A through its Laplace transform, and T uniquely determines its associated integral semigroup S (Lemma 6.5 and Definition 6.6) which uniquely determines K by (1.1). By Theorem 6.7, T (t) maps into the domain of A and AT (t)µ + µ = S(t)µ. Choosing µ = δx for x ∈ Ω yields the forward equation (1.5).


27

Theorem 7.3. Let K be a non-degenerate transition kernel and A the Hille-Yosida operator associated with K in Theorem 7.2. Then Z t K(t, x, ·) = δx + A dsK(s, x, ·) 0

for all t ≥ 0, x ∈ Ω. Further K is uniquely determined by this equation. Even the following holds: if K is a transition kernel which satisfies this equation and sup

k(t, x, Ω) < ∞

for all σ > 0,

0≤t≤σ,x∈Ω

then K is a Markov transition kernel, the operator family S(t)µ = R µ(dx)K(t, x, ·) is the integral semigroup generated by A, and the Ω Laplace transform of K is related to the resolvent of A as in Theorem 7.2. Proof. To see the last statement, we apply Proposition 6.11 with X = ˜ = BM(Ω), Y = {δx ; x ∈ Ω}, S(t) as just defined and M(Ω) and X Rt [T (t)µ](D) = 0 [S(r)µ](D)dr. ¤ ˜ =BM(Ω) can be identified with a closed subspace of X ∗ which X ˜ ∩ X ¯ is a total closed is invariant under T ∗ . By Proposition 6.4, X ˜ ∩ X¯ = X ˜ ◦ and the restricsubspace of X ∗ . By (2.6) and (7.3), X ˜ ∩ X ¯ coincides with S˜◦ , the restriction of S˜ to X ˜◦. tion of S ¯ to X ˜ ˜ ˜ Let R A◦ be the generator of S◦ . Then, for f ∈ D(A◦ ) and v(t)(x) = K(t, x, dy)f (y), x ∈ Ω, v is the unique solution of the backward Ω equation (1.7), v 0 = A˜◦ v, v(0) = f . The Markov transition function K ˜ be anis uniquely determined by the backward equation. Indeed, let K R ˜ other Markov transition function such that v˜(t)(x) = Ω K(t, x, dy)f (y) also solves the backward equation for all f ∈ D(A˜◦ ). By uniqueness, R R ˜ x, dy)f (y) for all f ∈ D(A˜◦ ). Since D(A˜◦ ) K(t, x, dy)f (y) = Ω K(t, Ω ˜ ◦ , this holds for all f ∈ X ˜ ◦ . Since X ˜◦ = X ˜ ∩ X ¯ is a total is dense in X ∗ ˜ x, ·). subspace of M(Ω) , K(t, x, ·) = K(t, Appendix A. A characterization of the Feller property Proposition A.1. Every f ∈ Cb (Ω) is the uniform limit of Pnfunctions Pn g = k=1 γk χCk and the uniform limit of functions h = k=1 γk χUk where 0 < γj < ∞ and (i) all Ck are closed Gδ -sets, (ii) all Uk are open Fσ -sets. If f ∈ C0 (Ω), one can arrange that (iii) all Ck are compact Gδ -sets,

28


(iv) all Uk are open Kσ -sets with compact closure. Proof. We can assume that f ∈ Cb (Ω) is non-negative. Then f can be uniformly approximated by functions n X

g(x) =

αk χ{αk−1 ≤f 0. Moreover Ck is a Gδ -set because Ck = ∞ m=1 {αk − m < f }. Alternatively f is the uniform limit of functions h of the form h(x) =

n X

αk χ{αk−1 0, there ˜ such that K(x, ˜ exists a compact subset C˜ of Ω C) < ² for all ˜ ˜ x ∈ Ω \ C. Condition (iii) is also necessary. Condition (i) and (ii) are necessary as well if Ω is locally compact. Proof. We first notice that upper and lower semi-continuity are preserved under uniform limits of functions. By Proposition A.1 (iii), f is the uniform limit of certain functions g which are positive linear combinations of χCk with compact Gδ -sets Ck . By (i) Z n X ˜ ˜ γk K(x, Ck ) g(y)K(x, dy) = Ω

k=1

is a linear combination (with positive scalars) of upper semi-continuous functions and so upper semi-continuous itself. Since f is the uniform R ˜ limit of functions g of this form, Ω f (y)K(x, dy) is the uniform limit of upper semi-continuous functions andR so upper semi-continuous itself. ˜ By (ii) and Proposition A.1 (iv), Ω h(y)K(x, dy) is the linear combination (with positive scalars) of lower semi-continuous functions and so lower semi-continuous itself. Since f is the uniform limit of funcR ˜ dy) is the uniform limit of lower semitions of the form h, Ω f (y)K(·, continuous functions and so lower semi-continuous itself. We have R ˜ dy) is both lower and upper semi-continuous. shown that Ω f (y)K(·, Hence it is continuous. Now let ² > 0. Then there exists some compact subset C of Ω such that f (x) < 2² for all x ∈ Ω \ C. By (iii), there exists a compact subset ² ˜ such that K(x, ˜ ˜ \ C. ˜ So, for all C˜ of Ω C) < 2(1+kf for all x ∈ Ω k) ˜ \ C, ˜ x∈Ω Z Z Z ˜ ˜ ˜ f (y)K(x, dy) = f (y)K(x, dy) + f (y)K(x, dy) Ω

C

Ω\C

² ˜ ≤kf kK(x, C) + < ². 2

Now let f ∈ C0 (Ω) be arbitrary. Then f = f+ − f− with non-negative ˜ by our previous f+ , f− ∈ C0 (Ω). Then S(f ) = S(f+ ) − S(f− ) ∈ C0 (Ω) consideration.

30


To see the necessity of (iii), let C be a compact subset of Ω. Then there exists a continuous nonnegative function f with compact support R ˜ dy) ∈ C0 (Ω). ˜ Let in Ω such that f ≥ χC . By assumption Ω f (y)K(·, ˜ such that ² > 0. Then there exists a compact subset C˜ of Ω Z ˜ ˜ ≤ ˜ ˜ \ C. ˜ K(x, C) f (y)K(x, dy) < ² ∀x ∈ Ω Ω

We now assume that Ω is locally compact. To show the necessity of (i), let C be a compact Gδ set in Ω. Then χC is the pointwise limit of a monotone decreasing sequence of continuous function fn on Ω with compact support. By the theorem of dominated convergence, Z ˜ ˜ K(x, C) = lim fn (y)K(x, dy). n→∞

Ω

˜ By assumption, K(C, ·) is the pointwise limit of a monotone decreas˜ ing sequence of continuous functions. Hence K(C, ·) is upper semicontinuous [12, III.10.4]. To show the necessity of (ii), let U be an open Kσ -set. Then χU is the pointwise limit of a monotone increasing sequence of continuous functions fn on Ω with compact support. A similar argument as before ˜ U ) is lower semi-continuous. shows that K(·, ¤ A similar proof provides the following result. Proposition A.3. Let Ω be a Hausdorff topological space. Then S ˜ if the following three conditions hold: maps Cb (Ω) into Cb (Ω) ˜ C) is upper semi(i): For every closed Gδ Baire set C in Ω, K(·, ˜ continuous on Ω. ˜ U ) is lower semi(ii): For every open Fσ Baire set U in Ω, K(·, ˜ continuous on Ω. The conditions (i) and (ii) are also necessary if Ω is normal. The necessity proof is based on Urysohn’s characterization of normality [12, Sec. VII.4]. Appendix B. Joint measurability Let (Θ, Σ) be a measurable space and Ω be a normal topological space with the σ-algebra B of Baire sets. Let [0, τ ) be equipped with the standard topology and the associated σ-algebra Bτ of Borel sets. Proposition B.1. Let Ω be normal and f : [0, τ ) × Θ → Ω such that f (t, θ) is a measurable function of x ∈ Θ for every t ∈ (0, τ ) and f (t, θ) : [0, τ ) → Ω is a right continuous function of t ≥ 0 for all θ ∈ Θ.


31

Then f : ([0, τ ) × Θ, Bτ × Σ) → (Ω, B) is measurable. Proof. Since the Baire-σ-algebra on the normal space Ω is generated by open Fσ sets [3, 7.2.3], we can assume that D is S open and the union ˜ ˜ of an increasing sequence of closed set Cn , D = ∞ n=1 Cn , and show −1 that f (D) ⊆ Bτ × Σ. By Urysohn’s lemma, there exist continuous functions gn : Ω → [0, 1] such that gn (x) = 0 for x ∈ Ω \ D and gn (x) = 1 for x ∈ Cñ . Define g(x) =

∞ X

2−n gn (x),

x ∈ Ω.

n=1

Then g is continuous and D = {x ∈ Ω; g(x) > 0}. Set Un = {x ∈ 1 Ω; g(x) > n+1 } and Cn = {x ∈ Ω; g(x) ≥ n1 }. Then D=

[ n∈N

Cn =

[

Un ,

Cn ⊆ Un ⊆ Cn+1 .

n∈N

Claim 1: The following two statements are equivalent for t ∈ [0, τ ), ω ˜ ∈ Θ: (a) f (t, ω ˜ ) ∈ D. (b) There exists some n ∈ N such that for all k ∈ N there exists some q ∈ (0, τ ) ∩ Q ∩ [t, t + 1/k) such that f (q, ω ˜ ) ∈ Cn . ‘(a)⇒(b)’: Choose n ∈ N such that f (t, ω ˜ ) ∈ Cn . Since Cn ⊆ Un and Un is open and f (·, ω ˜ ) is right continuous, for all k ∈ N, there exists some q ∈ (0, τ ) ∩ Q ∩ [t, t + 1/k), such that f (q, ω ˜ ) ∈ Un ⊆ Cn+1 . This implies (b). ‘(b)⇒(a)’: Choose n ∈ N such that for all k ∈ N there exists q ∈ (0, τ ) ∩ Q ∩ [t, t + 1/k) with f (q, ω ˜ ) ∈ Cn . We can choose a sequence qk & t in (0, τ ) ∩ Q such that f (qk , ω ˜ ) ∈ Cn . Since f (·, ω ˜ ) is rightcontinuous and Cn is closed, f (t, ω ˜ ) ∈ Cn ⊆ D. Claim 1 can easily be rewritten in the following way. Claim 2: The following two statements are equivalent for ω ˜ ∈ Θ: (a) t ∈ [0, τ ), f (t, ω ˜ ) ∈ D. (b) (∃n ∈ N)(∀k ∈ N)(∃q ∈ (0, τ ) ∩ Q): t ∈ (q − 1/k, q] ∩ [0, τ ) and f (q, ω ˜ ) ∈ Cn . Claim 2 can be rewritten set-theoretically as

32


f

−1

¶ 1 i (D) = q − , q ∩ [0, τ ) k n∈N k∈N q∈Q∪(0,τ ) n o ˜ × ω ˜ ∈ Ω; f (q, ω ˜ ) ∈ Cn . [ \

[

µ³

Observe that the sets on the left of × are in Bτ . Since f (q, ·) is measurable for fixed q, and Cn ∈ B, the sets on the right of × are elements of Σ. Hence f −1 (D) ∈ Bτ × Σ, because countable intersections and unions of measurable sets are again measurable. ¤ Acknowledgement. The authors thank Doug Blount and Mats Gyllenberg for useful comments. References [1] W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel 2001 [2] L. Arnold, Stochastic Differential Equations: Theory and Applications, John Wiley, New York 1974 [3] H. Bauer, Probability Theory and Elements of Measure Theory, Academic Press, London 1981 [4] C.D. Aliprantis, Owen Burkinshaw, Principles of Real Analysis, third edition, Academic Press, San Diego 1998 ´ment, O. Diekmann, M. Gyllenberg, H.J.A.M. Heijmans, [5] Ph. Cle H.R. Thieme, A Hille-Yosida theorem for a class of weakly continuous semigroups, Semigroup Forum 38 (1989), 157-178 ´ment, H.J.A.M. Heijmans, S. Angenent, C.J. van Duijn, B. de [6] Ph. Cle Pagter, One-Parameter Semigroups, North-Holland, Amsterdam 1987 [7] E.B. Davies, One-Parameter Semigroups, Academic Press, London 1980 [8] M. Demuth, J.A. van Casteren, Stochastic Spectral Theory for Selfadjoint Feller Operators, Birkhäuser, Basel 2000 [9] O. Diekmann, M. Gyllenberg, J.A.J. Metz, H.R. Thieme, On the formulation and analysis of deterministic structured population models. I. Linear theory, J. Math. Biol. 43 (1998), 349-388 [10] O. Diekmann, M. Gyllenberg, H.R. Thieme (1993), Perturbing semigroups by solving Stieltjes renewal equations, Diff. Integ. Eqn. 6 (1993), 155181 [11] O. Diekmann, M. Gyllenberg, H.R. Thieme (1995), Perturbing evolutionary systems by step responses and cumulative outputs. Diff. Integ. Eqn. 8, 1205-1244 [12] J. Dugundji, Topology, Allyn and Bacon, Boston 1966, Prentice-Hall of India, New Delhi 1975 [13] N. Dunford, J.T. Schwartz, Linear Operators Part I: General Theory, John Wiley, New York 1988


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[14] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York 2000 [15] S.N. Ethier, T.G. Kurtz, Markov Processes. Characterization and Convergence, Wiley, New York 1986 [16] W. Feller, An Introduction to Probability Theory and its Applications, Vol II, Wiley 1965 [17] W. Feller, Semi-groups of transformations in general weak topologies, Annals Math. 57 (1953), 287-308 [18] A. Friedman, Stochastic Differential Equations and Applications, Vol. 1, Academic Press, New York 1975 [19] J.A. Goldstein, Semigroups of Linear Operators and Application, Oxford University Press, New York 1985 [20] M. Gyllenberg, T. Lant, H.R. Thieme, Perturbing dual evolutionary systems by cumulative outputs, Differential Integral Equations 19 (2006), 401436 [21] E. Hille, R.S. Phillips, Functional Analysis and Semi-Groups, AMS, Providence 1957 [22] T. Lant, Transition Kernels, Integral Semigroups on Spaces of Measures, and Perturbation by Cumulative Outputs, Dissertation, Arizona State University, Tempe, December 2004 [23] T. Lant, H.R. Thieme, Perturbation of transition functions and a FeynmanKac formula for the incorporation of mortality, Positivity, to appear [24] B. Øksendal, Stochastic Differential Equations, Springer, Berlin Heidelberg 1985 [25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York Berlin Heidelberg 1982 [26] K. Taira, Semigroups, Boundary Value Problems, and Markov Processes, Springer, Berlin Heidelberg 2004 [27] H.R. Thieme, Integrated semigroups and integral solutions to abstract Cauchy problems, J. Math. Anal. Appl. 152 (1990), 416-447. [28] H.R. Thieme, Positive perturbations of dual and integrated semigroups, Adv. Math. Sci. Appl. 6, 445-507, 1996 [29] H.R. Thieme, Balanced exponential growth for perturbed operator semigroups, Adv. Math. Sci. Appl. 10 (2000), 775-819 [30] H.R. Thieme, J. Voigt, Stochastic semigroups: their construction by perturbation and approximation, Proc. Positivity IV - Theory and Applications, (M.R. Weber, J. Voigt, eds.), 135-146, Technical University of Dresden, Dresden 2006 [31] J.A. van Casteren, Generators of strongly continuous semigroups, Pitman, Boston 1985 [32] J. van Neerven, The Adjoint of a Semigroup of Linear Operators, Lecture Notes in Mathematics 1529, Springer, Berlin Heidelberg 1992

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T. Lant, H.R. Thieme †

present address: Decision Center for a Desert City, Arizona State University, PO Box 878209, Tempe, AZ 85287-8209, U.S.A. E-mail address: [email protected] †¦

Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804, U.S.A. E-mail address: [email protected]