Ergodic measures of Markov semigroups with thee–property

Tomasz Szarek∗, Daniel T.H. Worm†

== MI-2010-09 ==

April 12, 2010

Abstract

We study the set of ergodic measures for a Markov semigroup on a Polish statespace. The principal assumption on this semigroup is the e–property, an equicon-tinuity condition. We introduce a weak concentrating condition around a compactset K and show that this condition has several implications on the set of ergodicmeasures, one of them being the existence of a Borel subset K0 of K with a bijectivemap from K0 to the ergodic measures, by sending a point in K0 to the weak limitof the Cesaro averages of the Dirac measure on this point. We also give sufficientconditions for the set of ergodic measures to be countable and finite. Finally, we givea quite general condition under which the Cesaro averages of any measure convergeto an invariant measure.

1 Introduction

In this paper we are concerned with the study of ergodicity of Markov semigroups. Litera-ture devoted to ergodic properties of Markov semigroups is huge. As a basic introductionmay serve the monograph by S. P. Meyn and R. L. Tweedie [14]. Since its publication in1993 there has been a rapid progress caused by possible applications in stochastic differen-tial equations and fractals. In the theory of stochastic differential equations for instance,it was usually assumed that Markov processes corresponding to solutions of studied equa-tions satisfy the strong Feller property. Since it is a very restrictive assumption, especially

∗University of Gdansk, ul. Wita Stwosza 57, 80-952 Gdansk, Poland†dworm@math.leidenuniv.nl, Mathematical Institute, University Leiden, P.O. Box 9512, 2300 RA Lei-

den, The Netherlands

1

in the case when noise is degenerate, there has been an urgent need to find out a new toolallowing us to examine degenerate stochastic differential equations. A first attempt in thisdirection was made by M. Hairer and J. Mattingly who introduced the so-called asymptoticstrong Feller property. Its definition is complex, hence omitted here. The reader interestedin it is referred to [6]. On the other hand, the main assumption we have made is the e–property. For the first time it was used by A. Lasota and one of the authors in [12], wherea sufficient condition for the existence of an invariant measure was formulated and proved.Let (U(t))t≥0 be a Markov semigroup defined on the class of all bounded Borel measurablefunctions on some Polish space S. We will say that this semigroup has the e-property ifthe family (U(t)ϕ)t≥0 is equicontinuous for any bounded Lipschitz function ϕ. The crite-rion mentioned above says that every semigroup with the e-property which is concentratedaround some compact set, admits an invariant measure. Further results on semigroupssatisfying the e-property are proved in [11]. The authors formulated criteria assuring theexistence of an invariant measure and its uniqueness. They also applied them to a stochas-tic differential equation corresponding to a passive tracer model. Recently the criteria forthe existence and uniqueness of an invariant measure have been extended to other stochas-tic differential equations – equations driven by Levy noise among others [10]. It is worthmentioning here that the solution to the 2D Navier–Stokes equation with degenerate noiseconsidered by Hairer and Mattingly generates the semigroup with the e-property not onlywith the asymptotic strong Feller property. It seems that all known examples of Markovprocesses with the asymptotic strong Feller property satisfy the e-property as well. Thisgives an additional reason for studying Markov semigroups with the e-property. RecentlyS.C. Hille and the second author started considering ergodic decompositions of generalMarkov semigroups on Polish spaces. It appeared that then a quite general Yosida-typedecomposition of the state space holds [18]. Similar results were obtained by O. Costaand F. Dufour in [3] in the setting of locally compact separable metric spaces. One ofits consequences is a characterisation of ergodic measures in terms of a measurable subsetof the state space, and an integral decomposition over this subset of any invariant mea-sure in terms of the ergodic measures. In [19] S.C. Hille and the second author focusedon Markov semigroups with the e–property, and showed interesting consequences for theergodic decompositions, some of which will be used in this paper.

In the present paper we are interested in determining ergodic measures as limits of Cesaroaverages starting from some compact set. A concentrating condition related to the oneintroduced in [12] appears to be perfectly fitted to our task. Namely, we prove that thenthe number of invariant ergodic measures is closely related to the behaviour of Markovsemigroups on this concentrating compact set. This allows us to provide a condition forthe existence of finitely many ergodic measures. Similar problems for infinite dimensionalsystems were studied in [13]. We may also determine whether the Markov semigroupadmits countably or uncountably many ergodic invariant measures.

The present paper is organised as follows. We start with some notational conventions andpreliminaries. In Section 3 we first show some new consequences of the e–property on theset of ergodic measures, and the remainder is devoted to the study of conclusions we are able

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to draw from a weak concentrating (around some compact set) condition. The main result,Theorem 3.8, says that the set of all ergodic measures is obtained as Cesaro weak limitsstarting at some points from the given compact set. In fact, we find in this way a Borelsubset of the compact set that maps bijectively to the set of ergodic measures. This allowsus to determine how many ergodic measure do exist. The condition assuring the existenceof finitely or countably many ergodic measures is provided in Section 4. In Section 5, inturn, we show (Theorem 5.2) that a condition related to our weak concentrating conditionensures on a Markov semigroup with the e–property that for every probability measurethe Cesaro weak limit exists and is an invariant measure. This theorem implies corollariesthat give necessary and sufficient conditions for a Markov semigroup to be weak* meanergodic and asymptotically stable.

Some notational conventions. Unless otherwise mentioned, (S, d) will denote a com-plete separable metric space, viewed as a measurable space with respect to its Borel σ-algebra. We writeM(S) to denote the real vector space of all signed finite Borel measureson S, containing M+(S), the cone of positive measures. P(S) consists of the probabilitymeasures in M+(S). We denote the total variation norm on M(S) by ‖ · ‖TV and writeM(S)TV for the Banach space consisting ofM(S) endowed with the total variation norm.We write 11E for the indicator function of E ⊂ S. For f : S → R measurable and µ ∈M(S)we write 〈µ, f〉 for

∫Sf dµ. Cb(S) denotes the Banach space of bounded continuous func-

tions from S to R, endowed with the supremum norm ‖ · ‖∞, and BL(S) the Banach spaceof bounded Lipschitz functions from S to R, with the norm ‖f‖BL = |f |Lip + ‖f‖∞, where|f |Lip denotes the Lipschitz constant of f . For x ∈ S and r > 0, B(x, r) denotes the openball around x with radius r.

2 Preliminaries

Let (S, d) be a complete separable metric space. On M(S) we can consider the weaktopology σ(M(S), Cb(S)) (not to be confused with the weak topology on M(S) whenviewed as a Banach space with the total variation norm). On M+(S) this topology ismetrisable by the norm ‖ · ‖∗BL:

‖µ− ν‖∗BL = sup{|〈µ− ν, f〉| : f ∈ BL(S) : ‖f‖BL ≤ 1},

and M+(S) is actually complete with respect to this metric (see e.g. [5, Theorem 9 andTheorem 18]).

We will make use of the well known Alexandrov Theorem several times in this paper. Itstates that a sequence of probability measures (µn)n converges weakly to a probabilitymeasure µ iff lim infn→∞ µn(U) ≥ µ(U) for all open U ⊂ S The above conditions areequivalent to the condition: lim supn→∞ µn(C) ≤ µ(C) for all closed C ⊂ S.

We can define the Banach space SBL to be the completion ofM(S) in BL(S)∗, thenM+(S)is a closed convex cone in SBL. See [8] for some properties of this Banach space.

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We define a Markov semigroup (P (t))t≥0 on S to be semigroup of maps P (t) :M+(S)→M+(S), such that P (t) is a positively homogeneous and additive map and

‖P (t)µ‖TV = ‖µ‖TV for all µ ∈M+(S).

Throughout this paper we will assume that (P (t))t≥0 is Markov–Feller, i.e. there is a dualsemigroup (U(t))t≥0 on Cb(S) such that

〈P (t)µ, f〉 = 〈µ, U(t)f〉

for every t ∈ R+, µ ∈M+(S), f ∈ Cb(S). We also assume that (P (t))t≥0 is jointly measur-able, i.e. for every Borel set E ⊂ S the map (t, x) 7→ P (t)δx(E) is jointly measurable. Thejoint measurability implies that for f ∈ L1(R+),

∫R+f(s)P (s)µ(E) ds exists for all E ∈ Σ

and µ ∈ M+(S) and this defines a measure. We can also define this integral as Bochnerintegral in SBL: as shown in [9, Section 2] we obtain for all h ∈ BM(S):⟨∫

R+

f(s)P (s)µ ds, h

⟩=

∫R+

〈f(s)P (s)µ, h〉 ds.

We call a measure µ ∈ P (S) invariant if P (t)µ = µ for every t ∈ R+ and write Pinv(S) todenote the convex set of invariant probability measures.

A Borel set E is called µ-invariant if for all t ∈ R+, U(t)11E = 11E µ-a.e. Then an invariantprobability measure µ is ergodic if µ(E) = 0 or µ(E) = 1 for every µ-invariant Borel setE (see e.g. [4]). The ergodic measures are also exactly the extreme points of Pinv(S). Wedenote the set of ergodic measures by Perg(S).

For t > 0 and µ ∈M+(S) we can define

P (t)µ =1

t

∫ t

0

P (s)µ ds,

then P (t) defines a Markov–Feller operator.

WriteΓt := {x ∈ S : (P (t)δx)t≥1 is tight}

andΓcp := {x ∈ S : P (t)δx converges in SBL as t→ +∞}.

Obviously Γcp ⊂ Γt. If x ∈ Γcp, then we define εx = limt→+∞ P(t)δx. Clearly εx ∈ P (S).

Because (P (t))t≥0 is Markov–Feller, it is not hard to show that εx ∈ Pinv(S).

We furthermore defineΓcpie := {x ∈ Γcp : εx is ergodic}.

On Γcpie we can define an equivalence relation ∼: x ∼ y iff εx = εy. We shall write [x]to denote the equivalence class of x. In [17, 18] it is shown that Γt,Γcp and Γcpie and [x]

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are Borel sets and that µ(Γt) = µ(Γcp) = µ(Γcpie) for all invariant probability measures µ.Furthermore, Perg(S) = {εx : x ∈ Γcpie} and εx([x]) = 1 for all x ∈ Γcpie.

A semigroup (P (t))t≥0 has the e–property if for every f ∈ BL(S), the family of functions(U(t)f)t≥0 is equicontinuous.

The e–property has various important consequences: In [19] it was shown that Γcp and Γcpieare closed sets. Moreover, the map Φ : Γcp → SBL of the form Φ(x) = εx, is continuousand every invariant probability measure µ satisfies µ =

∫Γcpie

εx dµ(x).

From [10, Lemma 1] it follows that Γt = Γcp (see also [19, Theorem 5.13]).

Since Φ is continuous and Γcpie is closed, [x] = Φ−1({εx}) is closed in S for every x ∈ Γcpie,thus in particular supp(εx) ⊂ [x].

3 Weak concentrating condition

We will always assume that (P (t))t≥0 is a jointly measurable Markov–Feller semigroup ona complete separable metric space (S, d). We begin this section by giving some propertiesof the set of ergodic measures when the e–property holds.

Proposition 3.1. Suppose (P (t))t≥0 has the e–property. Then Perg(S) is closed in SBL.

Proof. An invariant probability measure µ is ergodic if and only if for every f ∈ Cb(S),limt→+∞ U

(t)f(x) = 〈µ, f〉 for µ-a.e. x ∈ S.

Let us define for f ∈ Cb(S)

f ∗(x) :=

{〈εx, f〉 if x ∈ Γcp,0 if x 6∈ Γcp.

For every x ∈ Γcp, U(t)f(x)→ 〈εx, f〉 = f ∗(x). Since µ(Γcp) = 1 we obtain that µ is ergodic

if and only if f ∗ = 〈µ, f〉 µ-.a.e. for every f ∈ Cb(S), or equivalently∫Γcp

(f ∗(x)− 〈µ, f〉)2 dµ(x) = 0. (1)

Now assume that (µn)n is a sequence of ergodic measures such that µn → µ in SBL. Thenµ is invariant, since (P (t))t≥0 is Markov–Feller. Fix f ∈ Cb(S). We need to show that (1)holds. Since ‖f ∗‖∞ ≤ ‖f‖∞, we have for every x ∈ Γcp and n ∈ N

|(f ∗(x)− 〈µ, f〉)2 − (f ∗(x)− 〈µn, f〉)2|≤ 2|f ∗(x)||〈µ− µn, f〉|+ |(〈µ, f〉)2 − (〈µn, f〉)2|≤ 2‖f‖∞|〈µ− µn, f〉|

+|〈µ+ µn, f〉||〈µ− µn, f〉|≤ 4‖f‖∞|〈µ− µn, f〉|.

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So ∣∣∣∣∣∫

Γcp

(f ∗(x)− 〈µ, f〉)2 − (f ∗(x)− 〈µn, f〉)2 dµn(x)

∣∣∣∣∣→ 0

as n→ +∞.

For every n ∈ N∣∣∣∣∣∫

Γcp

(f ∗(x)− 〈µ, f〉)2 dµ(x)−∫

Γcp

(f ∗(x)− 〈µnf〉)2 dµn(x)

∣∣∣∣∣ ≤∣∣∣∣∣∫

Γcp

(f ∗(x)− 〈µ, f〉)2 d[µ(x)− µn(x)]

∣∣∣∣∣ (2)

+

∣∣∣∣∣∫

Γcp

(f ∗(x)− 〈µ, f〉))2 − (f ∗(y)− 〈µn, f〉)2 dµn(x)

∣∣∣∣∣ .The final term in inequality (2) above goes to zero as n→ +∞.

Since x 7→ εx is continuous from Γcp to SBL, we also know that x 7→ (f ∗(x) − 〈µ, f〉)2

is bounded and continuous from Γcp to R. Γcp is closed, thus we can apply the TietzeExtension Theorem, and so there exists a g ∈ Cb(S), such that g(x) = (f ∗(x) − 〈µ, f〉)2

for every x ∈ Γcp. Since µ(Γcp) = µn(Γcp) = 1 for every n ∈ N, we have∣∣∣∣∣∫

Γcp

(f ∗(x)− 〈µ, f〉)2 d[µ(x)− µn(x)]

∣∣∣∣∣ = |〈µ, g〉 − 〈µn, g〉| → 0

as n→ +∞ since µn → µ in SBL and g ∈ Cb(S).

Now note that∫

Γcp(f ∗(x) − 〈µn, f〉)2 dµn(x) = 0 for every n ∈ N, since the measures µn

are ergodic, thus∫

Γcp(f ∗(x)− 〈µ, f〉)2 dµ(x) = 0 as well. Thus µ is ergodic.

Proposition 3.2. If Pinv(S) is non-empty, then there exists an invariant probability mea-sure µ0 with

supp(µ0) =⋃

µ∈Pinv(S)

supp(µ).

Hence⋃µ∈Pinv(S) supp(µ) is closed.

Proof. Let D =⋃µ∈Pinv(S) supp(µ). Then D is separable, so there exist (xn)n ⊂ D such

that D ⊂ {xn : n ∈ N}. Let µn ∈ Pinv(S) be such that xn ∈ supp(µn) and define

µ0 =∞∑n=1

(1/2n)µn,

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then µ0 ∈ Pinv(S) and∞⋃n=1

supp(µn) ⊂ supp(µ0),

thus

D =∞⋃n=1

supp(µn) = supp(µ0).

We can also ask ourselves what we can say about the union of the supports of all ergodicmeasures. Even with the e–property, this set need not be closed, as the following exampleshows:

Example 3.3. Let S = [0, 1]. For x ∈ S and t ∈ R+ define

P (t)δx = [x+ e−t(1− x)]δx + [(1− x)− e−t(1− x)]δ1−x.

Then P (0)δx = δx, and easy calculation shows that P (t)P (s)δx = P (t+ s)δx for all x ∈ Sand s, t ∈ R+. For every E ⊂ S Borel,

P (t)δx(E) = [x+ e−t(1− x)]11E(x) + [(1− x)− e−t(1− x)]11E(1− x)

so (t, x) 7→ P (t)δx(E) is jointly measurable. Thus we can define a jointly measurableMarkov semigroup on S as follows:

P (t)µ =

∫S

P (t)δx dµ(x) for all µ ∈M+(S).

It can be shown that (P (t))t≥0 is Markov-Feller and satisfies the e–property. Now, for allx ∈ S, P (t)δx → xδx + (1 − x)δ1−x = εx as t → ∞. Because these measures cannot bewritten as the convex combination of different invariant probability measures, these areergodic measures. So Γcpie = S, and thus each ergodic measure equals xδx+(1−x)δ1−x forsome x ∈ S. Now, for all 0 < x < 1, supp(εx) = {x, 1−x}, and supp(ε0) = supp(ε1) = {1}.Thus ⋃

x∈S

supp(εx) = (0, 1],

which is open but not closed in S.

We show that in general the e–property implies that union of the supports of ergodicmeasures is a Gδ subset of S, i.e. a countable intersection of open sets.

Theorem 3.4. Let (P (t))t≥0 be a Markov–Feller semigroup with the e-property. Then

D :=⋃

µ∈Perg(S)

supp(µ)

is an Gδ set. In particular, D is a Polish space in its relative topology.

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Proof. If x ∈ supp(µ) for an ergodic measure µ, then x ∈ Γcpie, and supp(µ) ⊂ [x], soµ = εx. So we can write

D = {x ∈ Γcpie : x ∈ supp(εx)} =⋂k∈N

Dk,

whereDk = {x ∈ Γcpie : εx(B(x, 1/k)) > 0}.

Let Ek := Γcpie\Dk = {x ∈ Γcpie : εx(B(x, 1/k)) = 0}. We will show that Ek is closed. Letxn ∈ Ek such that xn → x in S. Then x ∈ Γcpie and εxn → εx.

For N ∈ N define VN = B(x, 1/k) ∩ (∩n≥NB(xn, 1/k)) . Let y ∈ VN and define r :=sup{d(y, xn) : n ≥ N}. Since d(y, x) < 1/k and xn → x, r < 1/k, which implies that VNis open in S. Now, for all N ∈ N,

εx(VN) ≤ lim infn→+∞

εxn(VN) ≤ lim infn→+∞

εxn(B(xn, 1/k)) = 0.

Since VN ⊂ VN+1 for all N and ∪NVN = B(x, 1/k), εx(B(x, 1/k)) = 0 and Ek is closed.

We can write:D = Γcpie ∩

⋂k∈N

(S\Ek).

Since Γcpie is closed, D is a Gδ set.

The final statement follows from [1, Theorem 3.1.2], which states that every Gδ subset ofa Polish space is again a Polish space.

We introduce the weak concentrating condition:

(C) There exists a compact K ⊂ S such that for every ε > 0 and every x ∈ S

lim supt→+∞

P (t)δx(Kε) > 0,

where Kε = {x ∈ S : d(x,K) < ε}.

It turns out that we can obtain every ergodic measure from K:

Lemma 3.5. Suppose (C) is satisfied. For every x ∈ Γcpie, K ∩ supp(εx) 6= ∅.

Proof. Suppose Γcpie is non-empty and let x ∈ Γcpie such that K ∩ supp(εx) = ∅. Sincesupp(εx) is closed and K is compact, there exists an ε > 0 such that Kε ∩ supp(εx) = ∅.Thus εx(K

ε) = 0. In particular, εx(Kε/2) = 0. Let y ∈ supp(εx) ∩ [x], which is non-emptysince εx([x]) = 1, then

lim supt→+∞

P (t)δy(Kε/2) ≤ εx(Kε/2) = 0,

which contradicts (C).

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Note that (C) is a stronger condition than condition (E) considered in [12]. Thus [12,Theorem 3.1] implies that there exists an invariant measure when (C) is satisfied and(P (t))t≥0 satisfies the e–property. Then there must also exist an ergodic measure, and thusΓcpie and Γcp are non-empty.

We shall write K := K ∩ Γcpie. Since Γcpie is closed, K is compact. By Lemma 3.5,

Φ(K) = Perg(S), and by continuity of Φ we can conclude:

Corollary 3.6. If the e–property and (C) hold for (P (t))t≥0, then Perg(S) is compact inSBL.

The following result can be found in [2, Corollary 6.9.18]:

Proposition 3.7. Let X be a Polish space and R an equivalence relation on X with closedequivalence classes. If R(Z) ⊂ X is a Borel set for every closed Z ⊂ X, then R admits aBorel section, i.e. there is a Borel set B ⊂ X such that B contains exactly one element ofevery equivalent class.

Theorem 3.8. If (P (t))t≥0 satisfies the e–property and (C), then there exists a Borel setK0 ⊂ K such that

(i) x ∈ supp(εx) for all x ∈ K0. In particular K0 ⊂ Γcpie.

(ii) If x, y ∈ K0 with x 6= y, then εx 6= εy.

(iii) For every ergodic measure µ there is an x ∈ K0 such that µ = εx.

Proof. Let

X :=⋃

µ∈Perg(S)

supp(µ) ∩K,

then X is a Gδ set by Theorem 3.4, hence a Polish space in its relative topology by [1,Theorem 3.1.2]. Also, X ⊂ Γcpie.

Let us define an equivalence relation R on X as follows: xRy if and only if x and y arein the support of the same ergodic measure, so if and only if εx = εy. Note that xRy ifand only if x ∈ supp(εy) if and only if y ∈ supp(εx). Note that R is the restriction toX of the equivalence class ∼ on Γcpie we introduced earlier. For every x ∈ X, R(x) =supp(εx)∩K = supp(εx)∩X, thus R(x) is closed in X. In order to apply Proposition 3.7,we need to show that R(Z) is closed for all closed subsets Z in X.

Let Z be closed in X. Let Z be its closure in S, then Z = Z ∩ X. Furthermore, Z is aclosed subset of K ∩ Γcpie, hence compact. We claim that

R(Z) = {x ∈ Γcpie ∩K : εx(Z1/n) > 0 for all n ∈ N} =: WZ .

Let x ∈ R(Z), then there is a z ∈ Z such that x ∈ supp(εz), thus also z ∈ supp(εx). Hence,for all n ∈ N, εx(Z

1/n) ≥ εx(B(z, 1/n)) > 0, so x ∈ WZ .

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Now let x ∈ WZ . Suppose that supp(εx) ∩ Z = ∅, then compactness of Z implies thatthere is an n ∈ N such that εx(Z

1/n) = 0, which is a contradiction. So there is a z ∈ Zsuch that z ∈ supp(εx). Since Z ⊂ K, z ∈ Z ∩X = Z. Since z ∈ supp εx, x ∈ supp εz, sox ∈ R(Z).

Now it remains to show that WZ is Borel in X. We can write

WZ =⋂n∈N

W nZ ,

where W nZ = {x ∈ Γcpie ∩K : εx(Z

1/n) > 0}. Let

V nZ := (Γcpie ∩K)\W n

Z = {x ∈ Γcpie ∩K : εx(Z1/n) = 0},

and let xk ∈ V nz such that xk → x ∈ S. Then x ∈ Γcpie ∩K and

εx(Z1/n) ≤ lim inf

k→+∞εxk(Z

1/n) = 0,

thus x ∈ V nz . So V n

z is closed, and thus W nz is open (in the relative topology on Γcpie ∩K).

Then W nz ∩X is open in X, so

R(Z) = WZ = X ∩WZ =⋂n∈N

W nZ ∩X

is a Gδ subset of X, thus Borel. Application of Proposition 3.7 yields the existence of aBorel set K0 ⊂ X ⊂ Γcpie ∩K such that for every x ∈ X there is exactly one y ∈ K0 suchthat y ∈ R(x).

Thus (i) and (ii) are satisfied. Now let µ be an ergodic measure, then there is an x ∈ Γcpiewith µ = εx. Lemma 3.5 implies that there is a z ∈ supp(εx) ∩K ⊂ X and thus there isexactly one y ∈ K0 such that y ∈ R(z). Consequently εy = εz = εx = µ. This concludesthe proof.

Note that the set K0 from Theorem 3.8 need not be unique. For instance, if we let Sbe the unit circle and P (t)δx := δe2πitx, then (P (t))t≥0 defines a Markov–Feller semigroupwith the e–property with a unique ergodic measure given by the Lebesgue measure on S.Obviously we can choose K0 = {z} for any z ∈ S.

Theorem 3.8 raises the interesting question if for Markov–Feller semigroups with the e–property, but without (C), a result analogous to Theorem 3.8 holds.

We say that (P (t))t≥0 is sweeping from some family A of Borel subsets of S if

limt→∞

P (t)µ(A) = 0,

for all µ ∈ P (S) and A ∈ A.

The following result generalises [16, Proposition 3]:

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Proposition 3.9. Let (P (t))t≥0 be a Markov-Feller semigroup that satisfies the e–propertyand (C), then (P (t))t≥0 is sweeping from compact sets disjoint from Γt.

Proof. We define by K0 ⊂ K the set of all x ∈ K such that for any ε > 0

lim supt→+∞

P (t)µ(B(x, ε)) > 0 (3)

for some µ ∈ P(S). Observe that K0 is closed, hence compact. Now we show that (C)is also satisfied with K replaced with K0. Fix ε > 0. Then for each x ∈ K0 we definerx = ε/2. For x ∈ K\K0 there exists an rx > 0 such that limt→+∞ P

(t)µ(B(x, 2rx)) = 0for all µ ∈ P(S). By compactness of K there exist m,n ∈ N0, x1, ..., xm ∈ K0 andxm+1, ..., xn ∈ K\K0 such that K ⊂ ∪ni=1B(xi, rxi). Let r := min{rxi : 1 ≤ i ≤ n}, thenr > 0 and

Kr ⊂ ∪mi=1B(xi, ε) ∪(∪ni=m+1B(xi, 2rxi)

)⊂ Kε

0 ∪(∪ni=m+1B(xi, 2rxi)

).

Now,

lim supt→+∞

P (t)δx(Kε0) = lim sup

t→+∞P (t)δx(K

ε0) +

m∑k=n+1

lim supt→+∞

P (t)δx(B(xi, 2rxi))

≥ lim supt→+∞

P (t)δx(Kr) > 0.

[19, Proposition 6.1] yields that K0 ⊂ Γt = Γcp.

Suppose there is a compact L such that L ∩ Γt = ∅ and an α > 0 and µ ∈ P (S) such thatlim supt→∞ P (t)µ(L) > α. Since Γt is closed, there is an η > 0 such that Lη ∩ Γt = ∅.

We define

M := {ν ∈ P (S) : there exists γ < η such that lim inft→∞

P (t)ν(Γγt ) > 1− α/2}.

Note that Γt is non-empty and (P (t))t≥0-invariant (see [16, Lemma 1]), thus {δx : x ∈Γt} ⊂ M , and in particular M is non-empty as well. Also, M is convex and P (t)M ⊂ Mfor all t ∈ R+. From [16, Lemma 3] it follows that M is open in the weak topology, andsince K0 ⊂ Γt there is a σ > 0 such that whenever ν ∈ P (S) with supp(ν) ⊂ Kσ

0 , thenν ∈M .

Let x ∈ L, then (C) implies that there is a tx > 0 such that

αx := P (tx)δx(Kσ/20 ) > 0.

Since P (tx) is Markov-Feller, there is an rx > 0 such that P (tx)(δy(Kσ/20 )) > 0 for all

y ∈ B(x, rx). By compactness of L there exist x1, ..., xk ∈ L such that L ⊂ ∪ki=1B(xi, rxi).Define Θ = min1≤i≤k αxi/2 and

γ := sup{β ≥ 0 : P (t0)µ ≥ βν for some ν ∈M, t0 > 0}.

11

Now choose ν ∈M and t0 > 0 such that P (t0)µ ≥ βν holds with β > γ −Θα/(2k). Thenfor all t ≥ 0, P (t + t0)µ ≥ βP (t)ν and P (t)ν ⊂ M , thus we can choose ν ∈ M and t0 insuch a way that P (t0)µ(L) > α and ν(L) < α/2. Then

(P (t0)µ− βν)(L) ≥ α− α/2 = α/2.

So there exist j ∈ {1, ..., k} such that (P (t0)µ− βν)(B(xj, rxj)) ≥ α/(2k). Now

〈P (txj)(P (t0)µ− βν), 11Kσ0〉 = 〈P (t0)µ− βν, U(txj)11Kσ

0〉

=

∫S

P (txj)δx(Kσ0 )d[P (t0)µ− βν](x)

≥∫B(xj ,rxj )

P (txj)δx(Kσ0 )d[P (t0)µ− βν](x)

≥ Θα/(2k).

Set

ν =(P (txj + t0)µ− βP (txj)ν)(· ∩Kσ

0 )

(P (txj + t0)µ− βP (txj)ν)(Kσ0 )

,

then ν ∈M , and supp(ν) ⊂ Kσ0 . Let

ν = β(β + Θα/(2k))−1P (txj)ν + Θα/(2k)(β + Θα/(2k))−1ν.

Since P (txj)ν and ν are in M , ν is in M as well, since M is convex. Furthermore,

P (txj + t0)µ ≥ (β + Θα/(2k))ν,

which contradicts the fact that γ < β + Θα/(2k). This completes the proof.

4 Countably many ergodic measures

Let (P (t))t≥0 be a jointly measurable Markov–Feller semigroup with dual(U(t))t≥0. In this section we give some sufficient conditions for the set of ergodic measuresto be countable or finite.

First we note that the e–property, even when combined with the weak concentrating con-dition (C), does not guarantee that the set of ergodic measures is countable. A trivialexample is given by S = [0, 1] and (P (t))t≥0 the identity semigroup. Then δx is an ergodicmeasure for all x ∈ S.

Hairer and Mattingly introduced in [6] the asymptotic strong Feller property, which gen-eralises the well-known strong Feller property, and used it in combination with other con-ditions to show uniqueness of invariant measures. They give a sufficient condition for aMarkov semigroup to be asymptotic strong Feller in [6, Proposition 3.12]. However, this

12

condition makes sense only for Hilbert spaces. We give a more general condition that worksfor Polish spaces as well. We will show that this implies that there are at most countablymany ergodic measures, and when combined with (C), the number of ergodic measures isfinite.

We fix an x0 ∈ S. For f : S → R and θ > 0 we define the local Lipschitz constant

|f |Lip,θ := sup

{|f(x)− f(y)|

d(x, y): x 6= y;x, y ∈ B(x0, θ)

}.

We assume there are sequences tn ≥ 0 and δn ↓ 0 and a non-decreasing function C : R+ →R, such that for all f ∈ BL(S) and θ > 0

|U(tn)f |Lip,θ ≤ C(θ)[‖f‖∞ + δn|f |Lip]. (4)

Our next result gives lower bounds on distances between points in the supports of differentergodic measures. It generalises [7, Theorem 2.1] and its proof is based on the proof ofthat theorem. We include it here for completeness.

Proposition 4.1. Let µ and ν be ergodic measures and x ∈ supp(µ), y ∈ supp(ν). Then(4) implies that

d(x, y) ≥ 1

C(d(x, x0) ∨ d(y, x0)).

Proof. We define for n ∈ N the following metric on S:dn(x, y) = 1 ∧ ( 1√δnd(x, y)). These

metrics induce metrics on P(S) in the following way:

dn(µ, ν) = sup{|〈µ− ν, f〉| : f ∈ Lip1dn(S)},

whereLip1

dn(S) = {f : S → R : |f(x)− f(y)| ≤ dn(x, y) for all x, y ∈ S}.Then dn(µ, ν) ≤ 1 and limn→+∞ dn(µ, ν) = 1/2‖µ − ν‖TV, by [6, Lemma 3.4]. Notethat it suffices to only consider those f ∈ Lip1

dn(S) for which f(x0) = 0. For such f ,|f(x)| = |f(x)− f(x0)| ≤ dn(x, x0) ≤ 1, so ‖f‖∞ ≤ 1. Moreover

|fn(x)− fn(y)| ≤ 1√δnd(x, y) for all x, y ∈ S,

so |fn|Lip ≤ 1√δn

. Now we apply (4):

dn(P (tn)δx, P (tn)δy) ≤ sup{|U(tn)f(x)− U(tn)f(y)| : f ∈ Lip1dn(S) and f(x0) = 0}

≤ d(x, y)C(d(x, x0) ∨ d(y, x0))(1 +√δn).

Let µ1 and µ2 be two distinct ergodic measures, then they are mutually singular, so‖µ1 − µ2‖TV = 2. Suppose that there are x ∈ supp(µ1) and y ∈ supp(µ2) such that

13

d(x, y) < (C(d(x, x0) ∨ d(y, x0)))−1. Then we will show that ‖µ1 − µ2‖TV < 2, whichgives a contradiction. By assumption there is a Borel set E containing x and y such thatα := min(µ1(E), µ2(E)) > 0 and β := diam(E)C(d(x, x0) ∨ d(y, x0)) < 1. We can writeµi = ανi + (1− α)ρi, with νi, ρi ∈ P (S) and νi(E) = 1. Then

dn(µ1, µ2) = dn(P (tn)µ1, P (tn)µ2) ≤ αdn(P (tn)ν1, P (tn)ν2) + 1− α

≤ α

∫S

∫S

dn(P (tn)δw, P (tn)δz) dν1(w) dν2(z) + 1− α

≤ αβ(1 +√δn) + 1− α,

thus1/2‖µ1 − µ2‖TV ≤ 1− α(1− β) < 1,

which is a contradiction.

Corollary 4.2. Assume that (4) holds for some non-decreasing C : R+ → R. Then thereexist at most countably many ergodic measures.

Proof. We will show that for every bounded set B in S, there exist at most countablymany ergodic measures whose support intersects B. Since we can cover S with countablymany bounded sets, this proves that there are at most countably many ergodic measures.

Let B ⊂ S bounded, and define R := sup{d(x, x0) : x ∈ B} < ∞. Let µ be an ergodicmeasure with x ∈ supp(µ)∩B. Then Proposition 4.1 implies that for any ergodic measureν with µ 6= ν and y ∈ supp(ν) ∩B we have

d(x, y) ≥ 1/C(R). (5)

Now we choose for every ergodic measure µ with supp(µ) ∩ B 6= ∅ an x ∈ supp(µ) ∩ Band consider the open ball B(x, 1/(2C(R))). By (5) these balls are mutually disjoint.Separability of S implies that there can be only countably many of such balls, whichconcludes the proof.

Corollary 4.3. Assume that (4) holds for some non-decreasing C : R+ → R and thatcondition (C) holds. Then there are only finitely many ergodic measures.

Proof. Lemma 3.5 implies that the support of every ergodic measure has non-empty in-tersection with K. Since K is compact, it is bounded, so by the proof of Corollary 4.2there is an R > 0 such that whenever µ and ν are two distinct ergodic measures, thenB(x, 1/(2C(R)) ∩ B(y, 1/(2C(R))) = ∅ for every x ∈ supp(µ) ∩K and y ∈ supp(ν) ∩K.Since any subset of K is totally bounded, there can be only finitely number of mutuallydisjoint balls with radius (2C(R))−1 and center in K, so the number of ergodic measuresis finite as well.

14

Notice that the conditions in Corollary 4.3 not necessarily imply the existence of invariantmeasures. However, when combined with the e–property there do exist invariant measures.There exist examples of Markov–Feller semigroups with the e–property that do not satisfy(4) or even the asymptotic strong Feller property. At the beginning of this section we gavea trivial example of such a semigroup. See also [11, Remark 6]. As for now we do not knowany Markov–Feller semigroups that satisfy the asymptotic strong Feller property but notthe e–property. We now give a condition to ensure both properties:

Proposition 4.4. Suppose there exists a non-decreasing C : R+ → R such that for allt ∈ R+ and f ∈ BL(S)

|U(t)f |Lip,θ ≤ C(θ)[‖f‖∞ + |f |Lip] = C(θ)‖f‖BL.

Then (P (t))t≥0 satisfies the e–property. If in addition there is a function h : R+ → R+

such that limt→+∞ h(t) = 0 and

|U(t)f |Lip,θ ≤ C(θ)[‖f‖∞ + h(t)|f |Lip] for all t ∈ R+,

then (P (t)) satisfies (4) as well.

Proof. If xn → x ∈ S, then for all f ∈ BL(S),

supt≥0|U(t)f(xn)− U(t)f(x)| ≤ C(2d(x, x0))‖f‖BLd(xn, x)

for n large enough. This proves the e–property. It is clear that under the extra assumption,(P (t))t≥0 satisfies (4) as well.

5 Convergence of Cesaro averages

In this section we will formulate a condition on Markov–Feller semigroups with the e–property such that the Cesaro averages of all probability measures will converge weakly toinvariant measures.

Note that (C) is not sufficient. See [11, Remark 1] for an example of a Markov–Fellersemigroup (P (t))t≥0 having the e–property that satisfies an even stronger condition than(C), i.e. there is a z ∈ S such that

lim inft→+∞

P (t)δx(B(z, ε)) > 0

for all ε > 0 and all x ∈ S. However, as shown in [11, Remark 1], the set Γt for thissemigroup does not equal the whole space, and since Γcp = Γt, there exist probabilitymeasures for which the Cesaro averages do not converge.

It turns out that strengthening (C) by demanding a uniform lower bound depending on εwill give the result. We first prove a preliminary result:

15

Lemma 5.1. Let (P (t))t≥0 be a Markov–Feller semigroup that satisfies the e–property. LetK ⊂ S be compact. Then for all ε > 0 there exists a δ > 0 such that for every y ∈ K andx ∈ B(y, δ)

‖P (t)δx − P (t)δy‖∗BL < ε for all t ∈ R+.

Proof. Suppose that the statement does not hold. Then there exists an ε > 0, yn ∈ K, xn ∈B(yn, 1/n) and tn ∈ R+ such that

‖P (tn)δxn − P (tn)δyn‖∗BL ≥ ε.

There is a subsequence ynk such that ynk → y ∈ K and thus xnk → y.

The e–property and [19, Theorem 4.2] imply that the family of maps(P (t)δ·)t≥0 from S to SBL is equicontinuous. So

‖P (tnk )δynk − P(tnk )δy‖∗BL → 0

and‖P (tnk )δxnk − P

(tnk )δy‖∗BL → 0,

giving a contradiction.

Theorem 5.2. Let (P (t))t≥0 be a Markov–Feller semigroup that satisfies the e–property.Then the following two statements are equivalent:

(i) There exists a compact set K ⊂ S such that for every ε > 0 we may find α > 0 suchthat

lim supt→+∞

P (t)δx(Kε) ≥ α for x ∈ S. (6)

(ii) The set of ergodic measures is compact and (P (t)µ)t≥0 converges to an invariant mea-sure for every µ ∈ P(S).

Proof. (i)⇒(ii): By Corollary 3.6 the set of ergodic measures is compact in SBL.

It follows from [19, Theorem 5.13] that it is sufficient to show that (P (t)µ)t≥1 is tight forany µ ∈ P(S). By Cε we denote the family of all subsets of S who are contained in a finiteunion of closed ε–balls. Then tightness of (P (t)µ)t≥1 is equivalent to the following: for allε > 0 there is a C ∈ Cε such that lim inft→+∞ P

(t)µ(C) ≥ 1− ε [15, Lemma 3.2].

We first show that we can replace the lim sup condition by a lim inf condition.Step 1. There exists a compact set K ⊂ Γcp such that for every ε > 0 we may find β > 0such that

lim inft→+∞

P (t)δx(Kε) ≥ β for x ∈ S. (7)

16

Let a compact set K satisfying condition (6) be given. We define by K0 ⊂ K the set of allx ∈ K such that for any ε > 0

lim supt→+∞

P (t)µ(B(x, ε)) > 0 (8)

for some µ ∈ P(S). Observe that K0 is closed, hence compact. Similar as in the proof ofProposition 3.9 it follows that K0 satisfies (6) with K replaced with K0. [19, Proposition6.1] yields that K0 ⊂ Γt = Γcp, so P (t)δx converges for all x ∈ K0.

By compactness the set of ergodic measures is tight. So there exists a compact set K1 suchthat ν(K1) ≥ 1/2 for every ergodic measure ν. Now for an arbitrary invariant probabilitymeasure µ, we obtain

µ(K1) =

∫Γcpie

εx(K1) dµ(x) ≥ µ(Γcpie)/2 = 1/2.

Since every invariant measure is concentrated on the closed set Γcp, we may assume that

K1 ⊂ Γcp. Now we define the compact set K := K0 ∪K1 ⊂ Γcp.

Fix 0 < ε < 1/8. By Lemma 5.1 there exists a δ > 0 such that for all y ∈ K andx ∈ B(y, δ), ‖P (t)δx − P (t)δy‖∗BL < ε2.

Define g := (1 − εd(·, K)) ∨ 0, then |g|Lip ≤ 1/ε and ‖g‖∞ ≤ 1, so g ∈ BL(S) with

‖g‖BL ≤ 1/ε+ 1. Moreover, 1/211Kε/2 ≤ g ≤ 11Kε . Fix x ∈ Kδ and let y ∈ K be such thatd(x, y) < δ. Then

P (t)δx(Kε) ≥ 〈P (t)δx, g〉 ≥ 〈P (t)δy, g〉 − ‖P (t)δx − P (t)δy‖∗BL(1/ε+ 1)

≥ 1/2P (t)δy(Kε/2)− (ε+ ε2).

Since y ∈ Γcp, P(t)δy converges to the invariant probability measure εy, so we obtain

lim inft→+∞

P (t)δx(Kε) ≥ 1/2 lim inf

t→+∞P (t)δy(K

ε/2)− (ε+ ε2)

≥ 1/2εy(Kε/2)− (ε+ ε2)

≥ 1/4− ε− ε2 > 1/4− 2ε > 0.

Let α > 0 be such that (6) is satisfied with ε replaced with δ. Fix x ∈ S. Then there is aT > 0 such that P (T )δx(K

δ) ≥ α/2. Define

ρ :=P (T )δx(K

δ ∩ ·)P (T )δx(Kδ)

.

Then ρ ∈ P (S) and P (T )δx ≥ P (T )δx(Kδ)ρ ≥ (α/2)ρ.

By Fatou’s lemma

lim inft→+∞

P (t)ρ(Kε) ≥∫S

lim inft→+∞

P (t)δy(Kε) dρ(y) ≥ 1/4− 2ε.

17

Now,

lim inft→+∞

P (t)δx(Kε) = lim inf

t→+∞P (t)P (T )δx(K

ε) ≥ α/2(1/4− 2ε),

where the first equality follows from [11, Lemma 2]. Thus (7) is satisfied with β = α/2(1/4−2ε).

Step 2. For every ε > 0 there exists an open set U with K ⊂ U such that for all µ ∈ P (S)with µ(U) = 1 there is a C ∈ Cε for which

lim inft→+∞

P (t)µ(C) ≥ 1− ε.

Fix ε > 0 and let x ∈ K. Since (P (t)δx)t≥1 is tight, we may find Cx ∈ Cε and rx > 0 suchthat

lim inft→+∞

P (t)δy(Cx) ≥ 1− ε for all y ∈ B(x, rx).

Indeed, let Cx ∈ Cε/2 be such that

lim inft→+∞

P (t)δx(Cx) ≥ 1− ε/2.

Choose an arbitrary function f ∈ BL(S) such that 11Cx ≤ f ≤ 11Cx , where Cx = Cε/2x . Obvi-

ously, Cx ∈ Cε. By the e–property we may find rx > 0 such that |P (t)f(x)−P (t)f(y)| < ε/2for t ≥ 0 and y ∈ B(x, rx). Then

lim inft→+∞

P (t)δy(Cx) ≥ lim inft→+∞

U (t)f(y) ≥ lim inft→+∞

U (t)f(x)− ε/2

≥ lim inft→+∞

U (t)11Cx(x)− ε/2 ≥ 1− ε.(9)

Let {x1, . . . xN} ⊂ K be such that

K ⊂N⋃i=1

B(xi, rxi) := U.

Set C =⋃Ni=1Cxi and observe that C ∈ Cε. For µ ∈ P(S) with µ(U) = 1 we have by

Fatou’s lemma

lim inft→+∞

P (t)µ(C) ≥∫S

lim inft→+∞

P (t)δx(C) dµ(x) ≥ 1− ε. (10)

Step 3. For every ε > 0 and every µ ∈ P (S) there is a C ∈ Cε for which

lim inft→+∞

P (t)µ(C) ≥ 1− ε.

18

Let ε > 0 and µ ∈ P (S). Let U and C ∈ Cε be given by Step 2. Define

γ = sup{α ≥ 0 : ∃N, t1, . . . , tN ≥ 0, P (t1)P (t2) . . . P (tN )µ ≥ αν

ν ∈ P(S), lim inft→+∞

P (t)ν(C) ≥ 1− ε}.

We prove that γ = 1. Assume, contrary to our claim, that γ < 1.

Let ε > 0 be such that K ε ⊂ U . Let β ∈ (0, 1) be such that condition (7) holds with εreplaced with ε. If γ > 0, choose α ∈ ((γ − β)(1− β)−1, γ) ∩ [0, 1) and else choose α = 0.Then there exist N ∈ N, t1, . . . , tN ≥ 0, and ν ∈ P(S) such that

P (t1)P (t2) . . . P (tN )µ ≥ αν

andlim inft→+∞

P (t)ν(C) ≥ 1− ε.

Setµ = (1− α)−1(P (t1)P (t2) . . . P (tN )µ− αν)

and observe that µ ∈ P(S). Further, by (7) and Fatou’s lemma,

lim inft→+∞

P (t)µ(U) ≥ β,

so there is a t > 0 such thatP (t)µ(U) ≥ β/2.

Define

ν1 =P (t)µ(· ∩ U)

P (t)µ(U).

ThenP (t)µ = (1− α)−1(P (t)P (t1)P (t2) . . . P (tN )µ− αP (t)ν) ≥ (β/2)ν1

and hence

P (t)P (t1)P (t2) . . . P (tN )µ ≥ αP (t)ν + β/2(1− α)ν1

= (α + β/2(1− α))[(α + β/2(1− α))−1(αP (t)ν + β/2(1− α)ν1)].

Set ν2 = (α + β/2(1− α))−1(αP (t)ν + β/2(1− α)ν1). Observe that Step 2 implies that

lim inft→+∞

P (t)ν2(C) ≥ (α + β(1− α))−1[α lim inft→+∞

P (t)ν(C) + β(1− α) lim inft→+∞

P (t)ν1(C)]

≥ 1− ε.

19

Observation that α + β/2(1 − α) > γ leads to contradiction. Hence γ = 1. Further, forany α < 1 we find N and t1, . . . , tN > 0 such that

lim inft→+∞

P (t)µ(C) = lim inft→+∞

P (t)P (t1)P (t2) . . . P (tN )µ(C)

≥ α lim inft→+∞

P (t)ν(C) ≥ α(1− ε),

where the first equality follows from [11, Lemma 2]. Hence

lim inft→+∞

P (t)µ(C) ≥ 1− ε,

which completes the proof of implication (i)⇒(ii).

(ii)⇒(i): By compactness the set of ergodic measures is tight. So there exists a compactset K such that µ(K) ≥ 1/2 for all ergodic µ. Let ν be an invariant probability measure,then

ν(K) =

∫Γcpie

εx(K) dν(x) ≥ ν(Γcpie)/2 = 1/2.

Let x ∈ S. By assumption P (t)δx converges to an invariant probability measure ν, thus forall ε > 0,

lim inft→+∞

P (t)δx(Kε) ≥ ν(Kε) ≥ 1/2.

We call a Markov semigroup (P (t))t≥0 weak∗ mean ergodic if there exists a µ∗ ∈ P (S) suchthat

limt→+∞

P (t)µ = µ∗ for all µ ∈ P (S).

In [11, Theorem 2] there are given sufficient conditions for weak∗ mean ergodicity. We canuse Theorem 5.2 to give a necessary and sufficient condition for a Markov semigroup to beweak∗ mean ergodic.

Corollary 5.3. Let (P (t))t≥0 be a Markov–Feller semigroup that satisfies the e–property.Then the following are equivalent:

(i) (P (t))t≥0 is weak∗ mean ergodic.

(ii) There exists a z ∈ S such that for every ε > 0 we may find α > 0 such that

lim supt→+∞

P (t)δx(B(z, ε)) ≥ α for x ∈ S. (11)

Proof. The statement follows from Theorem 5.2 and the observation that if (ii) is satisfied,then condition (C) holds with K replaced with {z}, so Theorem 3.8 implies that there isexactly one ergodic measure.

20

We define (P (t))t≥0 to be asymptotically stable if there exists a probability measure µ∗ suchthat P (t)µ→ µ∗ for all µ ∈ P (S). Then µ∗ is a unique invariant probability measure.

Corollary 5.4. Let (P (t))t≥0 be a Markov–Feller semigroup with the e–property. Then(P (t))t≥0 is asymptotically stable if and only if there exists a z ∈ S such that for everyε > 0 we may find α > 0 for which

lim inft→+∞

P (t)δx(B(z, ε)) ≥ α for all x ∈ S. (12)

Proof. If (P (t))t≥0 is asymptotically stable with invariant probability measure µ∗, then forall z ∈ supp(µ∗) and all x ∈ S

lim inft→+∞

P (t)δx(B(z, ε)) ≥ µ∗(B(z, ε)) > 0.

Now suppose there is a z ∈ S such that (12) is satisfied. Then condition (C) holds with Kreplaced with {z}, so Theorem 3.8 implies that there is exactly one ergodic measure, henceone invariant probability measure µ∗. By [16, Theorem 2], P (t)µ → µ∗ for all µ ∈ P (S)with µ(Γt) = 1, and by Theorem 5.2 Γt = Γcp = S. This completes the proof.

21

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[19] D.T.H. Worm and S.C. Hille, Equicontinuous families of Markov operators on completeseparable metric spaces with applications to ergodic decompositions and existence,uniqueness and stability of invariant measures, submitted (2010).

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