introduction - arizona state universitythieme/transition06my3.pdf · 2007-05-11 · involve the...

MARKOV TRANSITION FUNCTIONS ANDSEMIGROUPS OF MEASURES

TIMOTHY LANT† AND HORST R. THIEME¦

Abstract. The application of operator semigroups to Markovprocesses is extended to Markov transition functions which do nothave the Feller property. Markov transition functions are char-acterized as solutions of forward and backward equations whichinvolve the generators of integrated semigroups and are shown toinduce integral semigroups on spaces of measures.

1. Introduction

Continuous semigroups of bounded linear operators have played animportant role as functional analytic tools in the theory of Markovprocesses [15, 16, 26]. The link is provided by (Markov) transitionfunctions K(t, x, D). Here x is a point in a state space Ω and D anelement of a σ-algebra B of subsets of Ω; t ≥ 0 is interpreted as timeand K(t, x, D) as the probability of the state being in the set D attime t provided x was the state at time 0. There are at least two waysof associating operator families with K (Section 2). The first type offamily operates on the Banach space of (signed) measures on B withbounded variation, M(Ω), and is defined by

(1.1) [S(t)µ](D) =

∫

Ω

µ(dx)K(t, x, D), µ ∈M(Ω),

[16, X.8], the second is the formally dual family on the Banach spaceof bounded measurable functions on Ω, BM(Ω), with supremum norm,

(1.2) [S(t)f ](x) =

∫

Ω

K(t, x, dy)f(y), f ∈ BM(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, Fellerproperty, spaces of measures.† partially supported by NSF grants DMS-0314529 and SES-0345945.¦ partially supported by NSF grants DMS-9706787 and DMS-0314529.

1

2 T. Lant, H.R. Thieme

[19, Sec.1.9]. The space of measures has an important interpretationin another application of Markov transition functions, structured pop-ulation models, as a positive measure µ represents the population dis-tribution with respect to the structure induced by the individual statespace Ω [9]. In this context, S(t)µ represents the structural distributionof the population at time t. Both operator families, S and S, are one-parameter semigroups. The semigroup property, S(t)S(r) = S(t + r)for all t, r ≥ 0, is equivalent to the Chapman-Kolmogorov equation

(1.3) K(t + r, x,D) =

∫

Ω

K(r, x, dy)K(t, y, D) ∀t, r ≥ 0.

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 norS(t)f are continuous in t in general. However, meaningful conditionsconcerning K can be found for S(t)f to be continuous in t if f isa continuous function on a suitable Hausdorff topological space, f ∈C(Ω). Let Ω be a compact metric space for the sake of exposition (wewill also look at the cases where Ω is a normal space or a locally compactspace). The continuity of S(t)f in t for f ∈ C(Ω) can most easily beexploited in a semigroup setting if S(t) leaves C(Ω) invariant such thatthe 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 asfollows (Sections 4 and A, cf. [31]):

(•) If D is an open set which is the countable union of compactsets (an open Kσ set) and t ≥ 0, then K(t, x, D) is a lowersemi-continuous function of x ∈ Ω. If D is a compact set whichis 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 sufficientto 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 metricspace, S is a dual semigroup on M(Ω) [5, 6, 28, 32] as M(Ω) can beidentified with the dual space of C(Ω).

The theory of operator semigroups has been so attractive to stochas-tic processes, because C0-semigroups have an infinitesimal generatorwhich allows the transition functions to be characterized as solutionsof two different types of operator differential equations, known as for-ward and backward equations. It is one of the purposes of this paper toget rid of the restriction (•) and still preserve these characterizations of

Transition functions and semigroups 3

transition functions. Other approaches can be found in [15, 17]. We usea more recent development in semigroup theory, integrated semigroups[1]. If we define

(1.4) [T (t)µ](D) =

∫ t

0

[S(r)µ](D)dr,

the operator family T (t) is a (once) integrated semigroup (Section6). 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., ifµ 6= 0 then T (t)µ 6= 0 for at least one t > 0. S is called the inte-gral semigroup associated with T (Section 6.1). Non-degeneracy holdsunder mild restrictions on K (Section 7) which follow from stochasticcontinuity (Section 3) if Ω is normal or σ-compact. A construction onBM(Ω) analogous to (1.4), which uses S instead of S, typically leads todegenerate integrated semigroups and is less fruitful. A non-degenerateintegrated semigroup is associated with a generator A, the generaliza-tion of the infinitesimal generator of a C0-semigroup. In our case, thegenerator A is a Hille-Yosida operator in M(Ω). The transition func-tion K is uniquely determined as the solution of the integral equation

(1.5) K(t, x, ·) = δx + A

∫ t

0

K(s, x, ·)ds, t ≥ 0, x ∈ Ω.

Here δx is the Dirac measure concentrated at x ∈ Ω. This equation canbe formally rewritten as a differential equation which corresponds tothe (Kolmogorov) forward equation (cf. [16, Sec.X.3 (3.5)], [15, Chap.4(9.58)]). In turn, A is uniquely determined by the transition functionas its resolvent can be expressed by the Laplace transform of K,

(λ− A)−1µ =

∫

Ω

µ(dx)

∫ ∞

0

dte−λtK(t, x, ·), µ ∈M(Ω).(1.6)

There is a subspace BM(Ω) of BM(Ω) which is total for M(Ω) andinvariant under the semigroup S such that the restrictions of S(t) toBM(Ω) form a C0-semigroup S. The infinitesimal generator of S,A, is dual to A and has the following property: If f ∈ D(A) andv(t)(x) =

∫Ω

K(t, x, dy)f(y), then v is the unique solution of

(1.7) v′ = Av, 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 bythis operator differential equation. For compact Ω, BM(Ω) contains


C(Ω) if the transition function K satisfies a sufficiently strong type ofstochastic 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 de-terministic models of structured populations, where this restriction isin the way of a satisfactory perturbation theory which would allowdeath 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 ona 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 hereoperate on a Banach space X and will satisfy the extra condition

(2.2) lim supt0

‖S(t)x‖ < ∞ ∀x ∈ X.

We call these semigroups B0-semigroups. It follows from the uniformboundedness principle and from (2.1) that any B0-semigroup is expo-nentially bounded, i.e., there exist M ≥ 1, ω ∈ R such that

‖S(t)‖ ≤ Meωt ∀t > 0.

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

(2.3) ‖S(t)x− x‖ → 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 stronglycontinuous in t ≥ 0. For C0-semigroups the infinitesimal generator Ais defined by

(2.5) Ax = limt0

(1/t)(S(t)x− x), 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; ‖S(t)x− x‖ → 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.


2.3. Transition functions. Let Ω and Ω be measurable spaces withrespective σ-algebras B and B.

Definition 2.1 ([3, 10.3.1]). A function K : Ω × B → R is called ameasure kernel if

(a) K(x, ·) is a non-negative measure on B for all x ∈ Ω.(b) K(·, D) is a B-measurable function for all D ∈ B.

A measure kernel K is called bounded if supx∈Ω K(x, Ω) < ∞.

Definition 2.2. A function K : R+×Ω×B → R is called a transitionfunction 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 itsatisfies the Chapman-Kolmogorov equations (1.3).

A transition function is called a transition kernel if K(·, D) is BR+×Bmeasurable where BR+ is the σ-algebra of Borel sets on R+ and BR+×Bis the product σ-algebra, in other words if K is a measure kernel.

Remark 2.1. Sometimes the term ‘transition function’ is used such thatthe Chapman-Kolmogorov equations are included [15]. We follow theuse in [26, Sec.3.2] and [18, Sec.2.1] though it may not be clear whetherthey use ‘Markov’ to highlight the Chapman-Kolmogorov equations orthe 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 ofthe 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 boundedlinear operators on the space of measures on B of bounded variation,M(Ω), and the space of bounded measurable functions, BM(Ω), respec-tively. The Chapman-Kolmogorov equations are equivalent to the semi-group property of these operator families. By property (d), ‖S(t)‖ ≤ c,‖S(t)‖ ≤ c for all t ∈ [0, δ] and both S and S are B0-semigroups andso exponentially bounded.

Lemma 2.3. There exist ω ∈ R, M ≥ 1 such that ‖S(t)‖ ≤ Meωt,‖S(t)‖ ≤ Meωt, K(t, x, Ω) ≤ Meωt for all t ≥ 0, x ∈ Ω.


Through

〈µ, f〉 =

∫

Ω

fdµ, f ∈ BM(Ω), µ ∈M(Ω),

BM(Ω) can be identified with a subspace of M(Ω)∗, the topologicaldual of M(Ω). Actually BM(Ω) is a sequentially weakly∗ closed sub-space which norms M(Ω), i.e. sup|〈f, µ〉|; ‖f‖ ≤ 1 is an equivalentnorm 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. SinceS and S are B0-semigroups on M(Ω) and BM(Ω) respectively, wecan use the construction (2.6) and obtain C0-semigroups S and S onM(Ω) and BM(Ω) respectively with infinitesimal generators A andA. These operators are in duality, as

〈Aµ, f〉 = 〈µ, Af〉 ∀µ ∈ D(A), f ∈ D(A).

The equations

(2.7) u′ = Au and v′ = Av

correspond to the (Kolmogorov) forward and backward equations. Thebackwards equation v′ = Av uniquely determines the transition func-tion K if BM(Ω) is a total subspace of M(Ω)∗, i.e. it separatesmeasures: if µ ∈ M(Ω) is not the 0 measure, then there exists somef ∈BM(Ω) such that 〈µ, f〉 6= 0. One of the tasks of this paper con-sists in finding practical conditions for this to be the case. The forwardequation in the form of u′ = Au is less useful because it seems difficultto come up with reasonable conditions which make M(Ω) separatepoints in BM(Ω). One of the reasons that this is difficult lies in thefact that an important class of Markov transition functions is of theform K(t, x, dy) = κ(t, x, y)dy, with Ω being a measurable subset ofRn. Then S(t) maps M(Ω) into the closed subspace of measures whichare Lebesgue absolutely continuous and which can be identified withL1(Ω). This implies thatM(Ω) is contained in L1(Ω). But L1(Ω) doesnot separate the zero function from functions which are 0 Lebesgue al-most everywhere. For the same class of Markov transition functions,the semigroup S is degenerate, i.e. if f is 0 almost everywhere (but noteverywhere), we still have that S(t)f is the zero-function in BM(Ω) forall t > 0. This makes S by itself not such a useful object of study and isone of the motivations to consider integrated and integral semigroupson M(Ω).


2.5. Feller’s warning. Feller notes in his classic [16, X.8] that semi-groups on M(Ω) seem to be a natural path to study Markov transitionfunctions but warns that general semigroups onM(Ω) may be too largea class to work with as not all of the semigroups are linked to transitionfunctions. The following holds, however.

Proposition 2.4. There is a bijective correspondance between Markovtransition functions and those B0-semigroups on M(Ω) the duals ofwhich leave BM(Ω) invariant.

Proof. One direction of the correspondence is clear from the previousconsiderations. 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) = 〈δx, S∗(t)χD〉 = 〈S(t)δx, χD〉 = [S(t)δx](D),

which shows that K(t, x, ·) is a measure on B. Let µ ∈M(Ω). Then

[S(t)µ](D) = 〈S(t)µ, χD〉 = 〈µ, S∗(t)χD〉 =

∫

Ω

µ(dx)K(t, x, D).

The other properties of a Markov transition function are easily checkedfor K. ¤

3. Stochastic continuity of transition functions

Little progress can be made in the interface of Markov transitionfunctions and one-parameter semigroups unless the state space Ω is atopological Hausdorff space. Exceptions are transition functions as-sociated with Markov jump processes [16, X.3] [15, 4.1] which induceC0-semigroups onM(Ω) when Ω is just a measurable space [30]. Recallthe 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 Banachspace of real-valued bounded continuous functions on Ω with supremumnorm. C0(Ω) denotes the set of continuous real-valued functions thatvanish at infinity. The latter means that for every ε > 0 there is acompact subset C of Ω such that |f(x)| < ε whenever x ∈ Ω \ C.

When Ω is a topological space, the following two σ-algebras are typ-ically considered: the σ-algebra of Borel sets generated by the opensubsets in Ω and the σ-algebra of Baire sets which is the smallest σ-algebra such that all continuous functions from Ω to R are measurable.In a topological space, we exclusively consider the σ-algebra of Baire


sets which is denoted by B. The reason is that we need all finite mea-sures on B to be regular in an appropriate sense; conditions which makeevery Borel measure regular seem to imply that the Borel and Bairesets 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 andevery 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 bythe open Fσ-sets [3, 7.2.3]. So, if every open set is an Fσ-set, the Baireand 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 setwith compact closure.

Definition 3.1. A measure kernel K is called weakly stochasticallycontinuous, if for every open Kσ-set U ⊆ Ω with compact closure

limt→0

K(t, x, U) = 1 whenever x ∈ U.

Definition 3.2. A function f : Ω → R is upper semi-continuous at apoint x0 in the topological space Ω if the set x ∈ Ω; f(x) < f(x0)+ εis open for every ε > 0. f is called lower semi-continuous at x0 if −fis upper semi-continuous at x0.

Theorem 3.3. The following are equivalent for a locally compact spaceΩ 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-continuousfunction 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 functionf ∈ 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 iscompact. 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 set U \D.We claim that U \D is a Kσ set. Since U is a Kσ-set, U =

⋃∞n=1 Cn for a


countable family of compact sets Cn. Since D is a Gδ-set, D =⋂n

k=1 Uk

for a countable family of open sets Uk. Now

U \D =U \ (n⋂

k=1

Uk) =∞⋃

k=1

(U \ Uk)

=∞⋃

k=1

(( ∞⋃n=1

Cn

)\ Uk

)=

∞⋃

k=1

( ∞⋃n=1

(Cn \ Uk)

).

So U \ D is the union of the countable family Cn \ Uk; k, n ∈ N ofcompact 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)χU ](x) is upper semi-continuous at t = 0

for every x ∈ Ω. Further, if C is a compact Gδ set, [S(t)χC ](x) is lowersemi-continuous at t = 0 for every x ∈ Ω. The same proof as inTheorem A.2 shows that [S(t)f ](x) is right continuous at t = 0. SinceS is a semigroup,

[S(t + h)f ](x) = [S(t)S(h)f ](x) =

∫

Ω

[S(h)f ](y)K(t, x, dy).

Since [S(h)f ](y) → f(y) as h 0, pointwise in x ∈ Ω, the dominatedconvergence 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

K(t, x, U) ≥∫

Ω

K(t, x, y)f(y)dy → f(x) = 1, t → 0.

Since U has compact closure, there exist some f ∈ C0(Ω) such thatf(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 isparacompact [12, XI.7] and every paracompact space is normal [12,VIII.2].

Corollary 3.4. Let Ω be σ-compact. Then, if the Markov transitionfunction K is weakly stochastically continuous, it is a transition kernel.


Proof. Let f ∈ C0(Ω). By Theorem 3.3, [S(t)f ](x) is right continu-ous in t ≥ 0 and Baire measurable in x ∈ Ω. By Proposition B.1,∫Ω

f(y)K(t, x, dy) is jointly Baire-measurable in (t, x). Let U be anopen Kσ-set. Then there exists a sequence (fn) in C0(Ω) such that fn 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 thatthe 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 totalsubspace of M(Ω)∗.

As we mentioned before, we have a greater interest in the semigroupS than in the semigroup S because the latter is degenerate for an im-portant class of Markov transition functions. We therefore reformulateTheorem 3.3 in terms of S using Lebesgue’s theorem of dominatedconvergence.

Corollary 3.6. The following are equivalent for a locally compact spaceΩ and a Markov transition function K:

(a) K is weakly stochastically continuous.(b) For all f ∈ C0(Ω) and µ ∈M(Ω), 〈f, S(t)µ〉 is a right-continu-

ous function of t ≥ 0.

3.2. General topological Hausdorff spaces.

Definition 3.7. A transition function K is called stochastically con-tinuous 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), 〈f, S(t)µ〉 is a right continuous function oft ≥ 0.

(b) If Ω is a completely regular space (e.g. a metric, normal, orlocally compact space), the stochastic continuity of K is equivalent tothe right continuity of 〈f, S(t)µ〉 in t for all f ∈ Cb(Ω), µ ∈M(Ω).

Proof. (a) This part can be proved similarly as for Theorem 3.3 andCorollary 3.6.

(b) Let x ∈ Ω be fixed but arbitrary. Let U 3 x be an open subsetof Ω. Since Ω is completely regular, there exists a continuous functionf : Ω → [0, 1] such that f(x) = 1 and f(y) = 0 for y ∈ Ω \ U . Then


f ∈ Cb(Ω) and f ≤ χU . Assume that 〈f, S(t)µ〉 is continuous in t ≥ 0for µ = δx, the Dirac measure concentrated at x. Then

K(t, x, U) ≥∫

Ω

f(y)K(t, x, dy) → 1, t → 0 + .

Since χΩ ∈ Cb(Ω),

0 ≤ K(t, x, U) ≤∫

Ω

χΩ(y)K(t, x, dy) → χΩ(x) = 1, t → 0 + .

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 atotal 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 bythe 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 disjointclosed sets. The proof proceeds now as the one of Corollary 3.4. ¤

4. Which transition functions induce C0-semigroups onC0(Ω)?

Often semigroup theory has been applied to Markov transition func-tions by assuming the Feller property, namely that the appropriatespace of continuous functions is invariant under S and a C0-semigroupis induced [16, X.10], [19, 9.11], [26, 3.2]. In this section, we clarify therestrictions that the Feller property imposes on the transition function.

Theorem 4.1. Let Ω be a locally compact Hausdorff space. Then aMarkov transition function induces a C0-semigroup on C0(Ω) if andonly 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) islower semi-continuous on Ω.

(iii): For every compact subset C of Ω, every t > 0, and every ε >

0, there exists a compact subset C of Ω such that K(t, x, C) < ε

for all x ∈ Ω \ C.(iv): K is weakly stochastically continuous.

A similar statement can be found in [31, Thm 2.1]. We are notable to prove the necessity-statement there (which has been left tothe reader) unless it is interpreted in the sense above. We need thefollowing abstract result [7, Prop.1.23][14, Ch.I, Thm.5.8].


Theorem 4.2. Let S(t); t ≥ 0 be a semigroup of bounded linearoperators on a Banach space X. Then S(·) is a C0-semigroup if (andonly if) it is weakly continuous at t = 0, i.e., 〈S(t)x, x∗〉 → 〈x, x∗〉 fort 0, x ∈ X, x∗ ∈ X∗.

Proof of Theorem 4.1. The necessity of (i), (ii), (iii) follows from Pro-position A.2. (Sufficiency:) By Proposition A.2, S(t) maps C0(Ω) intoitself 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 dominatedconvergence theorem that∫

Ω

[S¦(t)f ](x)µ(dx) →∫

Ω

f(x)µ(dx), t → 0,

for every non-negative Borel measure µ on Ω and also for every signedBorel measure of finite variation. Since C0(Ω)∗ can be identified withthe 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 Theorem4.2, S¦(·) is a C0-semigroup. ¤

Analogously one can characterize the Markov transition functionswhich induce C0-semigroups on Cb(Ω) for a normal space Ω by usingTheorem 3.8 and Proposition A.3.

Remark 4.1. By functional analytic magic (the Krein-Smulian theo-rem that the closed convex hull of a weakly compact set is weaklycompact [13, V.6.Thm.3]), the weak stochastic continuity of K (to-gether with the Feller property) implies the following locally uniformtime-continuity statements:

(i) Let C ⊂ U ⊂ Ω and C compact and U open. Then

lim inft→0

infx∈C

K(t, x, U) ≥ 1.

(ii) Let C ⊂ Ω and C compact. Then

lim supt→0

infx∈C

K(t, x, C) ≤ 1.

If Ω is a locally compact metric space, C a compact set in Ω andK(t, x, Ω) ≤ 1 for all t ≥ 0, x ∈ Ω, then it follows [26, Thm. 3.1] that

lim supt0

supx∈C

[1−K(t, x, Bε(x))

]= 0,

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


Proof. (i) Choose f ∈ C0(Ω) with f = 1 on C, f = 0 on U , and0 ≤ f(y) ≤ 1 for all y ∈ Ω. Since S induces a C0-semigroup on C0(Ω),

supx∈Ω

∣∣∣∫

Ω

K(t, x, dy)f(y)− f(x)∣∣∣ → 0, t → 0.

Let ε > 0. Then there exists some δ > 0 such that

f(x) + ε >

∫

Ω

K(t, x, dy)f(y) > f(x)− ε ∀t ∈ [0, δ].

For x ∈ C,

1− ε = f(x)− ε <

∫

U

K(t, x, dy)f(y) ≤ K(t, x, U).

Further, for x ∈ C,

1 + ε = f(x) + ε >

∫

C

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

¤

An obvious example of a Markov transition function which does nothave the Feller property is K(t, x, D) = e−tγ(x)δx(D) where γ is a non-negative Baire measurable function which is not continuous. Then theassociated semigroup on BM(Ω), [S(t)f ](x) = f(x)e−tγ(x), does notmap continuous functions f to continuous functions.

In the following example, with Ω = R+, the semigroup S preservescontinuity, but not the property of vanishing at infinity. We interpretexponential growth, N(t) = N0e

(β−µ)t, in an age-structured populationmodel: β and µ are the per capita birth and mortality rates, N(t) thepopulation size at time t and N0 the initial population size. We stratifythe 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

u(t, a) =

u0(a− t)e−µt; t < a,

βeβ(t−a)e−µt

∫ ∞

0

u0(s)ds; t > a,

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


measure) and S(t)u0 = u(t, ·). We use the formal duality 〈S(t)f, u0〉 =〈f, S(t)u0〉 to find

(4.2) [S(t)f ](a) = e−µt(f(a + t) +

∫ t

0

f(s)eβ(t−s)ds),

for f ∈ BM(R+). For t > 0, S(t) maps continuous functions to contin-uous functions, but lim

a→∞[S(t)f ](a) > 0 if f ∈ C0(R+) is positive. To

make this example rigorous, we observe that (4.2) defines a semigroupS on BM(R+) indeed and that the associated transition function is

K(t, a,D) = e−µt(δt+a(D) +

∫

[0,t]∩D

eβ(t−s)ds).

Similar problems arise in body-size structured population modelswhich involve per capita birth rates β(x) that depend on body sizex. Constructing the associated C0-semigroup [10] (strongly continuousevolutionary system [11]) on C0(R+) by perturbation requires β(x) → 0as x →∞, i.e. the birth rate must tend to 0 for large body sizes. Thisassumption which may be unrealistic in certain applications can bedropped if the Feller property does not need to be satisfied [20, 23].

5. More continuity results and a backward integralequation

Motivated by the characterization of the Feller property in the previ-ous section and its failure in the preceding examples, we want to workwithout it and the restrictions it imposes on Markov transition kernels.Recall the space BM(Ω) of those bounded measurable functions f onΩ for which S(t)f is a continuous function of t; the Markov transitionfunction 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 Ωis locally compact. This is of interest as the domain of infinitesimalgenerator A of S involved in the backward equation (1.7) (see alsoSection 2.4) is hard to characterize and one can alternatively considerthe integral version

(5.1) v(t) = f + A

∫ t

0

v(r)dr, f ∈ BM(Ω),

which is uniquely solved by v(t)(x) =∫

Ωf(y)K(t, x, dy).

Throughout this section, Ω is a locally compact Hausdorff space andK a Markov transition function. Further we assume that


(¦) for every ε > 0, x ∈ Ω, there exist an open neighborhood U ofx and δ > 0 such that

K(t, z, Ω) ≤ 1 + ε ∀z ∈ U, t ∈ [0, δ].

In many stochastic applications, condition (¦) is trivially satisfiedas K(t, x, Ω) ≤ 1. The perturbation theory in [23] leads to Markovtransition kernels which only satisfy (¦), but not this more restrictiveinequality. In the next results, we investigate a stronger stochasticcontinuity concept than the ones used in Section 3.

Theorem 5.1. C0(Ω) ⊆ BM(Ω) if and only if the following two state-ments hold:

(i) If U ⊂ Ω is an open Baire set, x ∈ U and ε > 0, then thereexist δ > 0 and an open set U 3 x such that

K(t, z, U) ≥ 1− ε whenever z ∈ U , 0 ≤ t < δ.

(ii) If C ⊆ Ω is a compact Baire set and ε > 0, then there exist acompact set C ⊆ Ω and δ > 0 such that

K(t, x, C) ≤ ε ∀x ∈ Ω \ C, 0 ≤ t < δ.

Corollary 5.2. Make the additional assumptions (i), and (ii) of The-orem 5.1. Then C0(Ω) ⊆ BM(Ω)0 and, for every f ∈ C0(Ω), the back-

ward integral equation v(t) = f + A0

∫ t

0v(r)dr has the unique continu-

ous solution v(t) =∫

ΩK(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 firstpart of the corollary follows directly from the previous theorem, theproperties of BM(Ω)0, and standard semigroup theory. The uniquenessof K follows from Proposition 3.5.

Proof of Theorem 5.1. ‘⇒’: Let U ⊆ Ω be open, x ∈ U and ε > 0.Since Ω is locally compact, there exist an open set U and a compactset 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 thatχC ≤ f ≤ χU . Then

K(t, x, C) ≤ [S(t)f ](x) → f(x), t 0,


uniformly in x ∈ Ω. Since f(x) = 0 for all x ∈ C, (ii) follows.‘⇐’: 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)fis not continuous at t = 0. Then there exist η > 0, a sequence tn 0and a sequence (xn) in Ω such that

(5.2)∣∣∣[S(tn)f ](xn)− f(xn)

∣∣∣ > 2η ∀n ∈ N.

We can assume that η ≤ 1.

Claim: There exists a compact set C ⊆ Ω such that xn ∈ C for alln ∈ N.

Suppose that the claim does not hold. By Definition 2.2 (d), thereexists 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. Sincef ∈ C0(Ω), there exists some compact set C1 ⊆ Ω such that

(5.3) |f(x)| ≤ η

8(1 + c)∀x ∈ Ω \ C1.

For all x ∈ Ω, t ≥ 0,

|S(t)(x)| ≤∫

C1

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

∫

Ω\C1

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

≤‖f‖K(t, x, C1) +η

8K(t, x, Ω).

(5.4)

By (ii), there exists a compact set C2 ⊆ Ω and some δ1 > 0 such that

(5.5) K(t, x, C1) ≤ η

8‖f‖+ 1x ∈ Ω \ C2, t ∈ [0, δ1].

Set δ = minδ1, 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 anxn we have∣∣[S(tn)f ](xn)− f(xn)

∣∣ ≤∣∣[S(tn)f ](xn)

∣∣+∣∣f(xn)

∣∣ ≤ η,

a contradiction. This proves the claim.After choosing subsequences, the real sequences (f(xn)) and ([S(tn)f ]

(xn)) converge. It follows from the claim that⋂∞

m=1 xn; n ≥ m 3 xfor 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

(5.6)∣∣[S(tn)f ](xn)− f(x)

∣∣ ≥ η ∀n ≥ mη.


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

∣∣[S(tn)](xn)−f(x)∣∣ ≤

∫

U

|f(y)− f(x)|K(tn, xn, dy)

+

∫

Ω\U|f(y)|K(tn, xn, dy) + |f(x)|[K(tn, xn, Ω)− 1

]+,

where [r]+ = maxr, 0 is the positive part of a real number r. By thedefinition of U ,

∣∣[S(tn)](xn)− f(x)∣∣ ≤η

8K(tn, xn, U) + ‖f‖[K(tn, xn, Ω)− 1

]+

+ ‖f‖(K(tn, xn, Ω)−K(tn, xn, U)).

Since every open set U ⊆ Ω with x ∈ U has non-empty intersectionwith all sets xn; n > m, m ∈ N, By (¦) and (i), there exists somen > mη such that

K(tn, xn, Ω) < 1 +η

8‖f‖+ 1≤ 2, K(tn, xn, U) > 1− η

8‖f‖+ 1.

For such an n > mη, we have

∣∣[S(tn)f ](xn)− f(x)∣∣ <

3η

4,

a contradiction to (5.6).¤

For a locally compact metric space, K is called locally uniformlystochastically continuous [26, 3.2] if for each δ > 0 and each compactsubset 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 5.1and 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 inTheorem 5.1.

(iii) =⇒ (i): Let U be an open set and x ∈ U . Then there existssome 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.


So K(t, x, Bη(x)) → 1 as t → 0+ uniformly on U . Let ε > 0. Thenthere exists some δ > 0 such that

K(t, x, U) ≥ K(t, x, Bη(x)) > 1− ε 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. Sleaves C0(Ω) invariant, but this property is not really used. In fact if itis satisfied, weak stochastic continuity instead of (iii) (or (i)) is sufficientin combination with (ii) to make the restriction of S to C0(Ω) a C0-semigroup (see Theorem 4.1). For further continuity results we referto [22].

6. Integrated semigroups

Our quest for a forward equation and a backward equation each ofwhich uniquely characterizes the Markov transition function leads us to(once) integrated semigroups, T , strongly continuous operator familieswhich satisfy

T (t)T (r) =

∫ t+r

0

T (s)ds−∫ t

0

T (s)ds−∫ r

0

T (s)ds, t, r ≥ 0,

T (0) = 0.

(6.1)

These relations can be motivated by formally defining T (t) =∫ t

0S(s)ds,

with a semigroup S. For an authoritative survey and bibliographicnotes 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 Aof a non-degenerate integrated semigroup is given as follows [27]: ifx, y ∈ X,

(6.2) x ∈ D(A), y = Ax ⇐⇒ T (t)x−tx =

∫ t

0

T (s)yds ∀t ≥ 0.

Notice that this definition makes sense and defines a closed operatorA, even if T is not an integrated semigroup. Actually one has thefollowing result:

Theorem 6.1. Let T (t), t ≥ 0, be a non-degenerate strongly continu-ous family of bounded linear operators on X and let the closed linear


operator A be defined by (6.2). Then T is an integrated semigroup if

and only if∫ t

0T (s)ds ∈ D(A) for all t ≥ 0 and

A

∫ t

0

T (s)xds = T (t)x− tx ∀t ≥ 0.

Proof. The “only if” part follows from [27, Lemma 3.4]. The “if” partfollows from the proof of [27, Thm.6.2]. ¤

If T (t) is exponentially bounded, i.e., there exist M, ω > 0 such that

‖T (t)‖ ≤ Meωt ∀t ≥ 0,

one has the following useful relation between the Laplace transform ofthe integrated semigroup and the resolvent of the generator. It followsby combining [1, Prop.3.2.4] and [27, Prop.3.10].

Theorem 6.2. Let T (t), t ≥ 0, be a strongly continuous exponentiallybounded family of bounded linear operators on X and A : D(A) →X be a linear operator in X. Then T is a non-degenerate integratedsemigroup and A its generator if and only if there exists some ω > 0such that any λ > ω is contained in the resolvent set of A and theresolvent of A can be expressed in terms of Laplace transforms of T ,

(6.3) (λ− A)−1 = λT (λ) := λ

∫ ∞

0

e−λtT (t)xdt.

Actually formula (6.3) can be used to define the generator A in thecase of exponentially bounded integrated semigroups [1, Def.3.2.1].

A particularly interesting family of (once) integrated semigroups arethose that are locally Lipschitz continuous (l.L.c.) [1, Sec.3.5].

Theorem 6.3. The following statements (i), (ii), and (iii) are equiv-alent for a closed linear operator A in a Banach space X:

(i) A is the generator of a non-degenerate integrated semigroup Tthat is l.L.c.: for any b > 0, there exists some Λ > 0 such that

‖T (t)− T (r)‖ ≤ Λ|t− r|, 0 ≤ r, t ≤ b.

(ii) A is the generator of a non-degenerate integrated semigroup Tand there exist constants M ≥ 1, ω ∈ R such that

‖T (t)− T (r)‖ ≤ M

∫ t

r

eωsds, 0 ≤ r ≤ t < ∞.

(iii) A is a Hille-Yosida operator, i.e., there exist M ≥ 1, ω ∈ Rsuch that (ω,∞) is contained in the resolvent set of A and

‖(λ− A)−n‖ ≤ M(λ− ω)−n, λ > ω, n = 1, 2, . . .


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) coin-cides with those x ∈ X for which T (t)x is continuously differentiable.

The derivatives S(t) = T ′(t)x, t ≥ 0, x ∈ D(A), provide bounded linear

operators S(t) from X = D(A) into itself forming a C0-semigroup onX which is generated by the part of A in X, A. Finally T (t) mapsX into X and

(2.10) T ′(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:

X =x ∈ X; ‖λ(λ− A)−1x− x‖ → 0, λ →∞=x ∈ X; ‖(1/h)T (h)x− x‖ → 0, h 0.(6.4)

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

(6.5) X¯ = x∗ ∈ X∗; ‖λ(λ− A)−1∗x∗ − x∗‖ → 0, λ →∞.If we want to emphasize the dependence of X¯ on the generator A, wewrite X¯

A . 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:

‖x‖ ≤ M sup|〈x, x¯〉|; x¯ ∈ X¯, ‖x¯‖ ≤ 1.Vice versa, X norms X¯. The restriction of (λ− A)−1∗ to X¯ formsa family of pseudoresolvents that is actually the resolvent of a closedlinear operator A¯ in X¯. It is easy to show that A¯ is densely definedin X¯ and, of course, a Hille-Yosida operator, and thus the infinites-imal generator of a C0-semigroup S¯ on X¯. We have the followingrelations:

X¯ =x∗ ∈ X∗; ‖(1/h)T ∗(h)x∗ − x∗‖ → 0, h 0

T ∗(t)x¯ =

∫ t

0

S¯(r)x¯dr, t ≥ 0, x¯ ∈ X¯,

〈S(t)x, x¯〉 =〈x, S¯(t)x¯〉, t ≥ 0, x ∈ X, x¯ ∈ X¯.

(6.6)

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


Then X® := X ∩ X¯ is a total subspace of X∗. Further the C0-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 ⇐⇒ 〈x,A®x¯〉 = 〈y, x¯〉 ∀x¯ ∈ D(A®).

Proof. Let x ∈ X and 〈x, x∗〉 = 0 for all x∗ ∈ X®. Let y∗ ∈ X.Since X is invariant under (λ − A)−1∗, (λ − A)−1∗y∗ ∈ X®. So 0 =〈x, (λ − A)−1∗y∗〉 = 〈(λ − A)−1x, y∗〉. Since this holds for all y ∈ X

and X is a total subspace of X∗, (λ − A)−1x = 0 and x = 0. Since(λ−A¯)−1 is the restriction of (λ−A)−1∗ to X¯, it leaves X® invariantand 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 handside of (6.7). Let y¯ ∈ X® and set

x¯ = (λ− A)−1∗y¯ = (λ− A®)−1y¯

for some sufficiently large λ > 0. Then x¯ ∈ D(A®) and

〈y, (λ− A)−1∗y¯〉 = 〈y, x¯〉 = 〈x,A¯x¯〉=〈x,A¯(λ− A¯)−1y¯〉 = 〈x,−y¯ + λ(λ− A¯)−1y¯〉.

By duality,

〈(λ− A)−1y, y¯〉 = 〈−x + λ(λ− A)−1x, y¯〉.Since y¯ ∈ X® has been arbitrary and X® is a total subspace of X∗,(λ− A)−1y = −x + λ(λ− A)−1x. Thus x ∈ D(A) and Ax = y. ¤6.1. Integral semigroups. We start with the following observationconcerning 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 linearoperators 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 satisfyingS(0)x = x for all x ∈ X. If T (t) is strongly continuous, then S isnon-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.


As T is non-degenerate, S(t)S(u) = S(t + u). Similarly we concludefrom T (r)S(0) = T (r) − T (0) = T (r) that S(0) = I. Now assumethat T (t), t ≥ 0, is strongly continuous, x ∈ X, and S(t)x = 0 for allt > 0. By (6.8), T (r)x is constant on every interval [t,∞), t > 0, andhence constant on (0,∞). By continuity and T (0) = 0 we have thatT (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 oper-ators 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 gener-ator of T , S is called the integral semigroup generated by A.

A closed linear operator A is called the generator of an integral semi-group if A generates a l.L.c. integrated semigroup T and there existsan integral semigroup S associated with T .

The integral semigroup S is uniquely determined by the generatorA because A uniquely determines T , and we will see later (Proposition3.6) that S uniquely determines A and, equivalently, T . The definitionof an integral semigroup also makes sense if the associated integratedsemigroup is not l.L.c. The following theorem also holds in this moregeneral 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 relationsfor 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 Theorem6.1,

T (r)S(t)x =T (r)x + A(∫ t+r

0

T (u)xdu−∫ r

0

T (u)xdu−∫ t

0

T (u)xdu)

= T (t + r)− T (t)x.

Hence (6.8) holds and S is the integral semigroup generated by A.


(i) ⇒ (ii): We have to show that T (t) maps into D(A). We use thedefinition in (6.2). By (6.8) and (6.1),

T (r)(S(t)x− x) = T (t + r)x− T (r)x− T (t)x =d

drT (r)T (t)x− T (t)x.

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 iscontinuously differentiable in t ≥ 0 and that S(t) = T ′(t), t ≥ 0, forma 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 continu-ous 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)−1S(t).

Proof. Let x ∈ X and S(t)x be continuous in t ≥ 0. Then, by theintegrated semigroup property,

T (r)

∫ t

0

S(u)xdu =

∫ t

0

(T (r + u)− T (u))xdu = T (r)T (t)x.

As T is non-degenerate,∫ t

0S(u)xdu = T (t)x. Hence T (t)x is continu-

ously differentiable and ddt

T (t)x = S(t)x. On the other hand, if T (t)x iscontinuously differentiable, then by the integrated semigroup property,

T (r)d

dtT (t)x = T (r + t)x− T (t)x = T (r)S(t)x.

As T is non-degenerate, ddt

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

〈x, S∗(t)x¯〉 = 〈S(t)x, x¯〉 = 〈x + AT (t)x, x¯〉

=〈x, x¯〉+ 〈x, T ∗(t)A¯x¯〉 = 〈x, x¯〉+⟨x,

∫ t

0

S¯(r)A¯xdr, x¯⟩

=〈x, S¯(t)x¯〉.¤

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


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∗ = limh0

(1/h) (S∗(h)x∗ − x∗). By Corollary 6.9,

(λ− A)−1∗y = limh0

(1/h)(S∗(h)− I)(λ− A)−1∗x∗

= limh0

(1/h)(S¯(h)− I)(λ− A)−1∗x∗

=A¯(λ− A)−1∗x∗ = −x∗ + λ(λ− A)−1∗x∗.

This implies that x∗ ∈ X¯ and y∗ = limh0(1/h)(S¯(h)x∗−x∗), hencex∗ ∈ D(A¯), y∗ = A¯x∗.

(b) By (a), A¯ is uniquely determined by S∗ and thus by S. SinceA¯ uniquely determines A by (6.7), A is uniquely determined by S. ¤

Proposition 6.11. Let X be a subspace of X∗ which separates points inX and Y a subset of X which separates points in X. Let S(t); t ≥ 0and T (t); t ≥ 0 be families of bounded linear operators such that S∗(t)and T ∗(t) map X into itself for all t ≥ 0. Assume that T (t) is stronglycontinuous in t ≥ 0 and 〈S(t)x, x∗〉 is Borel measurable in t ≥ 0 for allx ∈ X and x∗ ∈ X and

(6.9) 〈T (t)x, x∗〉 =

∫ t

0

〈S(r)x, x∗〉dr ∀x ∈ X, x∗ ∈ X.

Finally let A be a Hille-Yosida operator in X such that (λ−A)−1∗ mapsX into itself for all λ > ω and S(t)x = x+AT (t)x for all t ≥ 0, x ∈ Y .

Then T is an integrated semigroup and A its generator and S theassociated integral semigroup, further

〈(λ− A)−1x, x∗〉 =

∫ ∞

0

e−λt〈S(t)x, x∗〉dt, λ > ω, x ∈ X, x∗ ∈ X.

Proof. Recall Proposition 6.4. Let x¯ ∈ D(A®). By (6.7),

〈x, S∗(t)x¯〉 = 〈x, x¯〉+ 〈x, T ∗(t)A®x¯〉for all x ∈ Y . Since S∗(t)x¯ ∈ X® and T ∗(t)A®x¯ ∈ X® and Y

separates point in X ⊇ X®,

S∗(t)x¯ = x¯ + T ∗(t)A®x¯.

For all x ∈ X, by duality,

(6.10) 〈S(t)x, x¯〉 = 〈x, x¯〉+ 〈T (t)x,A®x¯〉.


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),

〈T (t)x, x¯〉 = t〈x, x¯〉+⟨∫ t

0

T (r)xdr,A®x¯⟩.

Again by Proposition 6.4, (6.7),

T (t)x = tx + A

∫ t

0

T (r)xdr.

Since A is a Hille-Yosida operator, it generates an integrated semi-group T (Theorem 6.3) and u(t) = T (t)x is the unique solution of

u(t) = tx + A∫ t

0u(r)dr [27, Thm.6.1]. This implies T = T and T is

an integrated semigroup and A its generator. By (6.11) and Theorem6.7, S is the integral semigroup associated with T . The relation be-tween the resolvent of A and the Laplace transform of S follows fromTheorem 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

(7.1) [T (t)µ](D) =

∫ t

0

(∫

Ω

µ(dx)K(s, x, D))ds, µ ∈M(Ω), D ∈ B.

Since K is jointly measurable, [S(t)f ](x) in (1.2) is jointly measur-able in (t, x) and we can change the order of integration ad libitum.This implies that T (·) is an integrated semigroup and S the associatedintegral semigroup,

(7.2) 〈f, T (t)µ〉 =

∫ t

0

〈f, S(r)µ〉dr ∀f ∈ BM(Ω), µ ∈M(Ω).

Further the dual integrated semigroup T ∗ leaves BM(Ω) invariant and

[T ∗(t)f ](x) =

∫ t

0

(∫

Ω

K(t, x, dy)f(y))ds =

∫ t

0

[S(s)f ](x)ds,

t ≥ 0, x ∈ Ω, f ∈ BM(Ω).

(7.3)

See [22] for details. We mention that there are one-parameter semi-groups S on M(Ω) (not induced by a Markov transition kernel) forwhich (1.4) does not lead to an integrated semigroup T [5, Sec.1].


7.1. Non-degeneracy. A transition function K is called non-degene-rate if a measure 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 derivetopological 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 tran-sition 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 Corollary3.9. Let µ ∈ M(B) and T (t)µ = 0 for all t ≥ 0. Since 〈f, S(t)µ〉 isright-continuous in t ≥ 0, it follows from (7.2) that 〈f, µ〉 = 0 for allf ∈ Cb(Ω) and µ = 0 because Cb(Ω) is a total subspace of M(Ω) byCorollary 3.9.

(b) The proof is similar to the one for (a) except that we use Propo-sition 3.4 and Proposition 3.5. ¤

7.2. Forward and backward equations. Let us now assume thatΩ is a measurable space with σ-algebra B and K is a non-degenerateMarkov transition kernel. Then the generator, A, of the associatednon-degenerate integrated semigroup, T , is defined and a Hille-Yosidaoperator, and the resolvents of A and the Laplace transform of T arerelated 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

[(λ− A)−1µ](D) =

∫

Ω

µ(dx)

∫ ∞

0

dte−λtK(t, x, D)

for all λ > ω, D ∈ B, µ ∈ M(Ω). Through this formula, K and Adetermine each other in a unique way.

Obviously A is uniquely determined by this equation. In turn, the in-tegrated semigroup T is uniquely determined by A through its Laplacetransform, and T uniquely determines its associated integral semigroupS (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).


Theorem 7.3. Let K be a non-degenerate transition kernel and A theHille-Yosida operator associated with K in Theorem 7.2. Then

K(t, x, ·) = δx + A

∫ t

0

dsK(s, x, ·)

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

sup0≤t≤σ,x∈Ω

k(t, x, Ω) < ∞ for all σ > 0,

then K is a Markov transition kernel, the operator family S(t)µ =∫Ω

µ(dx)K(t, x, ·) is the integral semigroup generated by A, and theLaplace transform of K is related to the resolvent of A as in Theorem7.2.

Proof. To see the last statement, we apply Proposition 6.11 with X =M(Ω) and X = BM(Ω), Y = δx; x ∈ Ω, S(t) as just defined and

[T (t)µ](D) =∫ t

0[S(r)µ](D)dr. ¤

X =BM(Ω) can be identified with a closed subspace of X∗ whichis invariant under T ∗. By Proposition 6.4, X ∩ X¯ is a total closedsubspace of X∗. By (2.6) and (7.3), X ∩ X¯ = X and the restric-tion of S¯ to X ∩ X¯ coincides with S, the restriction of S to X.Let A be the generator of S. Then, for f ∈ D(A) and v(t)(x) =∫Ω

K(t, x, dy)f(y), x ∈ Ω, v is the unique solution of the backward

equation (1.7), v′ = Av, v(0) = f . The Markov transition function K

is uniquely determined by the backward equation. Indeed, let K be an-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,∫Ω

K(t, x, dy)f(y) =∫

ΩK(t, x, dy)f(y) for all f ∈ D(A). Since D(A)

is dense in X, this holds for all f ∈ X. Since X = X ∩X¯ is a totalsubspace of M(Ω)∗, K(t, x, ·) = K(t, x, ·).

Appendix A. A characterization of the Feller property

Proposition A.1. Every f ∈ Cb(Ω) is the uniform limit of functionsg =

∑nk=1 γkχCk

and the uniform limit of functions h =∑n

k=1 γkχUk

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,


(iv) all Uk are open Kσ-sets with compact closure.

Proof. We can assume that f ∈ Cb(Ω) is non-negative. Then f can beuniformly approximated by functions

g(x) =n∑

k=1

αkχαk−1≤f<αk(x) + αnχαn≤f(x)

where 0 < α0 < · · · < αn are appropriately chosen numbers. g can berewritten as

g =n∑

k=1

αk

(χαk−1≤f − χαk≤f

)+ αnχαn≤f

=n−1∑

k=0

αk+1χαk≤f −n−1∑

k=1

αkχαk≤f

=α1χα0≤f +n−1∑

k=1

(αk+1 − αk)χαk≤f.

The sets Ck = αk ≤ f are closed and, if f ∈ C0(Ω), compact becauseαk > 0. Moreover Ck is a Gδ-set because Ck =

⋂∞m=1αk − 1

m< f.

Alternatively f is the uniform limit of functions h of the form

h(x) =n∑

k=1

αkχαk−1<f≤αk(x) + αnχαn<f(x),

0 < α0 < αk < αk+1 for k ∈ N. Similarly as before, we see thath is the linear combination (with positive scalars) of characteristicfunctions χUk

where Uk = αk < f. Uk is open and has compactclosure contained in the compact set αk ≤ f if f ∈ C0(Ω) be-cause αk > 0. It is also an Fσ-set (Kσ-set if f ∈ C0(Ω)), becauseUk =

⋃∞m=1αk + 1

m≤ f. ¤

Let Ω, Ω be Hausdorff topological spaces and B the σ-algebra of Bairesets in Ω.

Let K : Ω×B → R+ be a bounded measure kernel. We consider themap S :BM(Ω) → BM(Ω) defined by

(A.1) [S(f)](x) =

∫

Ω

K(x, dy)f(y), x ∈ Ω.

Proposition A.2. Let K be a bounded measure kernel. S maps C0(Ω)into C0(Ω) (in particular, K has the Feller property [26, p.56]) if thefollowing three conditions hold:


(i): For every compact Gδ Baire set C in Ω, K(·, C) is uppersemi-continuous on Ω.

(ii): For every open Kσ Baire set U in Ω with compact closure,K(·, U) is lower semi-continuous on Ω.

(iii): For every compact subset C of Ω and every ε > 0, thereexists a compact subset C of Ω such that K(x,C) < ε for allx ∈ Ω \ C.

Condition (iii) is also necessary. Condition (i) and (ii) are necessaryas well if Ω is locally compact.

Proof. We first notice that upper and lower semi-continuity are pre-served under uniform limits of functions. By Proposition A.1 (iii), fis the uniform limit of certain functions g which are positive linearcombinations of χCk

with compact Gδ-sets Ck. By (i)

∫

Ω

g(y)K(x, dy) =n∑

k=1

γkK(x,Ck)

is a linear combination (with positive scalars) of upper semi-continuousfunctions and so upper semi-continuous itself. Since f is the uniformlimit of functions g of this form,

∫Ω

f(y)K(x, dy) is the uniform limit ofupper semi-continuous functions and so upper semi-continuous itself.

By (ii) and Proposition A.1 (iv),∫

Ωh(y)K(x, dy) is the linear com-

bination (with positive scalars) of lower semi-continuous functions andso lower semi-continuous itself. Since f is the uniform limit of func-tions of the form h,

∫Ω

f(y)K(·, dy) is the uniform limit of lower semi-continuous functions and so lower semi-continuous itself. We haveshown that

∫Ω

f(y)K(·, dy) is both lower and upper semi-continuous.Hence it is continuous.

Now let ε > 0. Then there exists some compact subset C of Ω suchthat f(x) < ε

2for all x ∈ Ω \C. By (iii), there exists a compact subset

C of Ω such that K(x,C) < ε2(1+‖f‖) for all x ∈ Ω \ C. So, for all

x ∈ Ω \ C,∫

Ω

f(y)K(x, dy) =

∫

C

f(y)K(x, dy) +

∫

Ω\Cf(y)K(x, dy)

≤‖f‖K(x,C) +ε

2< ε.

Now let f ∈ C0(Ω) be arbitrary. Then f = f+ − f− with non-negativef+, f− ∈ C0(Ω). Then S(f) = S(f+)− S(f−) ∈ C0(Ω) by our previousconsideration.


To see the necessity of (iii), let C be a compact subset of Ω. Thenthere exists a continuous nonnegative function f with compact supportin Ω such that f ≥ χC . By assumption

∫Ω

f(y)K(·, dy) ∈ C0(Ω). Let

ε > 0. Then there exists a compact subset C of Ω such that

K(x, C) ≤∫

Ω

f(y)K(x, dy) < ε ∀x ∈ Ω \ C.

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 ofa monotone decreasing sequence of continuous function fn on Ω withcompact support. By the theorem of dominated convergence,

K(x,C) = limn→∞

∫

Ω

fn(y)K(x, dy).

By assumption, K(C, ·) is the pointwise limit of a monotone decreas-ing sequence of continuous functions. Hence K(C, ·) is upper semi-continuous [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 continuousfunctions fn on Ω with compact support. A similar argument as beforeshows that K(·, U) is lower semi-continuous.

¤A similar proof provides the following result.

Proposition A.3. Let Ω be a Hausdorff topological space. Then Smaps Cb(Ω) into Cb(Ω) if the following three conditions hold:

(i): For every closed Gδ Baire set C in Ω, K(·, C) is upper semi-continuous on Ω.

(ii): For every open Fσ Baire set U in Ω, K(·, U) is lower semi-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 nor-mality [12, Sec. VII.4].

Appendix B. Joint measurability

Let (Θ, Σ) be a measurable space and Ω be a normal topologicalspace with the σ-algebra B of Baire sets. Let [0, τ) be equipped withthe standard topology and the associated σ-algebra Bτ of Borel sets.

Proposition B.1. Let Ω be normal and f : [0, τ) × Θ → Ω suchthat f(t, θ) is a measurable function of x ∈ Θ for every t ∈ (0, τ) andf(t, θ) : [0, τ) → Ω is a right continuous function of t ≥ 0 for all θ ∈ Θ.


Then f : ([0, τ)×Θ,Bτ × Σ) → (Ω,B) is measurable.

Proof. Since the Baire-σ-algebra on the normal space Ω is generatedby open Fσ sets [3, 7.2.3], we can assume that D is open and the unionof an increasing sequence of closed set Cn, D =

⋃∞n=1 Cn, and show

that f−1(D) ⊆ Bτ × Σ. By Urysohn’s lemma, there exist continuousfunctions gn : Ω → [0, 1] such that gn(x) = 0 for x ∈ Ω \ D andgn(x) = 1 for x ∈ Cn. Define

g(x) =∞∑

n=1

2−ngn(x), x ∈ Ω.

Then g is continuous and D = x ∈ Ω; g(x) > 0. Set Un = x ∈Ω; g(x) > 1

n+1 and Cn = x ∈ Ω; g(x) ≥ 1

n. Then

D =⋃

n∈NCn =

⋃

n∈NUn, Cn ⊆ Un ⊆ Cn+1.

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 andUn is open and f(·, ω) is right continuous, for all k ∈ N, there existssome 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 sequenceqk t in (0, τ) ∩ Q such that f(qk, ω) ∈ Cn. Since f(·, ω) is right-continuous 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


f−1(D) =⋃

n∈N

⋂

k∈N

⋃

q∈Q∪(0,τ)

((q − 1

k, q

]∩ [0, τ)

)

×

ω ∈ Ω; f(q, ω) ∈ Cn

.

Observe that the sets on the left of × are in Bτ . Since f(q, ·) is mea-surable for fixed q, and Cn ∈ B, the sets on the right of × are elementsof Σ. Hence f−1(D) ∈ Bτ × Σ, because countable intersections andunions of measurable sets are again measurable.

¤

Acknowledgement. The authors thank Doug Blount and Mats Gyl-lenberg for useful comments.

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[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, LectureNotes in Mathematics 1529, Springer, Berlin Heidelberg 1992


† present address: Decision Center for a Desert City, ArizonaState University, PO Box 878209, Tempe, AZ 85287-8209, U.S.A.

E-mail address: [email protected]

†¦Department of Mathematics and Statistics, Arizona State Univer-sity, Tempe, AZ 85287-1804, U.S.A.

E-mail address: [email protected]

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