Relation between equicontinuity, the so called e property and stability of
Markov operators is studied. In particular, it is shown that any asymptotically
stable
  Markov operator with an invariant measure such that the interior of its
support is nonempty satisfies the e property

Hille, Sander C.

Szarek, Tomasz

Ziemlanska, Maria A.

Comptes Rendus Mathematique

English

arXiv

Sander Cornelis Hille

Tomasz Szarek

Maria Aleksandra Ziemlańska

Crossref

Equicontinuous families of Markov operators in view of asymptotic stability

Hille, Sander Cornelis

Ziemlańska, Maria Aleksandra

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 355 (2017) 1247–1251Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comFunctional analysis/Probability theoryEquicontinuous families of Markov operators in view of asymptotic stability ✩Familles équicontinues d’opérateurs markoviens du point de vue de la stabilité asymptotiqueSander Cornelis Hille a, Tomasz Szarek b, Maria Aleksandra Ziemlańska aa Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlandsb Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Polanda r t i c l e i n f o a b s t r a c tArticle history:Received 29 August 2017Accepted 30 October 2017Available online 15 November 2017Presented by the Editorial BoardThe relation between the equicontinuity – the so-called e-property – and the stability of Markov operators is studied. In particular, it is shown that any asymptotically stable Markov operator with an invariant measure such that the interior of its support is non-empty satisfies the e-property.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous étudions la relation entre l’équicontinuite – la dite e-propriété – et la stabilité d’opérateurs de Markov. En particulier, nous montrons que tout opérateur markovien asymptotiquement stable, avec une mesure invariante telle que l’intérieur de son support est non vide, satisfait la e-propriété.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionThis paper is centred around two concepts of equicontinuity for Markov operators defined on probability measures on Polish spaces: the e-property and the Cesàro e-property. Both appeared as a condition (among others) in the study of the ergodicity of Markov operators. In particular, they are very useful in proving the existence of a unique invariant measure and its asymptotic stability: at whatever probability measure one starts, the iterates under the Markov operator will weakly converge to the invariant measure. The first concept appeared in [8,12], while the second was introduced in [14] as a ✩ The work of Maria Aleksandra Ziemlańska has been partially supported by a Huygens Fellowship of Leiden University. The work of Tomasz Szarek has been supported by the National Science Centre of Poland, grant number 2016/21/B/ST1/00033.E-mail addresses: shille@math.leidenuniv.nl (S.C. Hille), szarek@intertele.pl (T. Szarek), m.a.ziemlanska@math.leidenuniv.nl (M.A. Ziemlańska).https://doi.org/10.1016/j.crma.2017.10.0191631-073X/© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1248 S.C. Hille et al. / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 1247–1251theoretical generalisation of the first and allowed the author to extend various results by replacing the e-property condition by the apparently weaker Cesàro e-property condition, among others.An interest in equicontinuous families of Markov operators existed already before the introduction of the e-property. Jamison [6], working on compact metric state spaces, introduced the concepts of (dual) Markov operators on the continuous functions that are ‘uniformly stable’ or ‘uniformly stable in mean’ to obtain a kind of asymptotic stability results in this setting. Meyn and Tweedie [10] introduced the so-called ‘e-chains’ on locally compact Hausdorff topological state spaces, for similar purposes. See also [15] for results in a locally compact metric setting.The above-mentioned concepts were used for proving ergodicity for some Markov chains (see [11,1–3,5,7,13]).It is worth mentioning here that similar concepts appear in the study of mean equicontinuous dynamical systems mainly on compact spaces (see, for instance, [9]). However, it must be stressed here that our space of Borel probability measures defined on some Polish space is non-compact.Studying the e-property, the natural question arose whether any asymptotically stable Markov operator satisfies this property. Proposition 6.4.2 in [10] says that this holds when the phase space is compact. In particular, the authors showed that the stronger e-chain property is satisfied. Unfortunately, the proof contains a gap, and it is quite easy to construct an example showing that some additional assumptions must be then added.On the other hand, striving to repair the gap of the Meyn–Tweedie result mentioned above, we show that any asymp-totically stable Markov operator with an invariant measure such that the interior of its support is nonempty satisfies the e-property.2. PreliminariesLet (S, d) be a Polish space. By B(x, r), we denote the open ball in (S, d) of radius r, centred at x ∈ S , and ∂ B(x, r) denotes its boundary. Further E , IntS E denote the closure of E ⊂ S and the interior of E , respectively. By Cb(S), we denote the vector space of all bounded real-valued continuous functions on S and by Bb(S) all bounded real-valued Borel measurable functions, both equipped with the supremum norm | · |. By Lb(S) we denote the subspace of Cb(S) of all bounded Lipschitz functions (for the metric d on S). For f ∈ Lb(S), Lip f denotes the Lipschitz constant of f .By M(S), we denote the family of all finite Borel measures on S and by P(S) the subfamily of all probability measures in M(S). For μ ∈M(S), its support is the setsuppμ := {x ∈ S : μ(B(x, r)) > 0 for all r > 0}.An operator P :M(S) →M(S) is called a Markov operator (on S) if it satisfies the following two conditions:(i) (positive linearity) P (λ1μ1 + λ2μ2) = λ1 Pμ1 + λ2 Pμ2 for λ1, λ2 ≥ 0; μ1, μ2 ∈M(S);(ii) (preservation of the norm) Pμ(S) = μ(S) for μ ∈M(S).A measure μ∗ is called invariant if Pμ∗ = μ∗ . A Markov operator P is asymptotically stable if there exists a unique invariant measure μ∗ ∈P(S) such that Pnμ → μ∗ weakly as n → ∞ for every μ ∈P(S).For brevity, we shall use the notation:〈 f ,μ〉 :=∫Sf (x)μ(dx) for f ∈ Bb(S), μ ∈ M(S).A Markov operator P is regular if there exists a linear operator U : Bb(S) → Bb(S) such that〈 f , Pμ〉 = 〈U f ,μ〉 for all f ∈ Bb(S), μ ∈ M(S).The operator U is called the dual operator of P . A regular Markov operator is a Feller operator if its dual operator U maps Cb(S) into itself. Equivalently, P is Feller if it is continuous in the weak topology (cf. [14], Proposition 3.2.2).A Feller operator P satisfies the e-property at z ∈ S if, for any f ∈ Lb(S), we havelimx→z lim supn→∞|Un f (x) − Un f (z)| = 0, (1)i.e. if the family of iterates {Un f : n ∈ N} is equicontinuous at z ∈ S . We say that a Feller operator satisfies the e-property if it satisfies it at any z ∈ S .D. Worm slightly generalized the e-property, introducing the Cesàro e-property (see [14]). Namely, a Feller operator Pwill satisfy the Cesàro e-property at z ∈ S if, for any f ∈ Lb(S), we havelimx→z lim supn→∞∣∣∣∣∣1nn∑k=1Uk f (x) − 1nn∑k=1Uk f (z)∣∣∣∣∣ = 0. (2)Analogously, a Feller operator satisfies the Cesàro e-property if it satisfies this property at any z ∈ S .The following simple example shows that Proposition 6.4.2 in [10] fails.S.C. Hille et al. / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 1247–1251 1249Example 2.1. Let S = {1/n : n ≥ 1} ∪ {0} and let T : S → S be given by the following formula:T (0) = T (1) = 0 and T (1/n) = 1/(n − 1) for n ≥ 2.The operator P :M(S) → M(S) given by the formula Pμ = T∗(μ) (the pushforward measure) is asymptotically stable, but it does not satisfy the e-property at 0.Jamison [6] introduced, for a Markov operator, the property of uniform stability in mean when {Un f : n ∈N} is an equicon-tinuous family of functions in the space of real-valued continuous functions C(S) for every f ∈ C(S). Here C is a compact metric space. Since the space of bounded Lipschitz functions is dense for the uniform norm in the space of bounded uniform continuous functions, this property coincides with the Cesàro e-property for compact metric spaces. Now, if the Markov operator P on the compact metric space is asymptotically stable, with the invariant measure μ∗ ∈ M1, then 1n∑ni=1 U i f → 〈 f , μ∗〉 pointwise, for every f ∈ C(S). According to Theorem 2.3 in [6], this implies that P is uniformly stable in mean, i.e. it has the Cesàro e-property.Example 2.2. Let (kn)n≥1 be an increasing sequence of prime numbers. SetS := {(kin−1−times︷ ︸︸ ︷0, . . . ,0, i/kn,0, . . .) ∈ l∞ : i ∈ {0, . . . ,kn},n ∈N}.The set S endowed with the l∞-norm ‖ · ‖∞ is a (noncompact) Polish space. Define T : S → S by the formulaT ((0, . . .)) = T ((kknn −1−times︷ ︸︸ ︷0, . . . ,0, 1,0, . . .)) = (0, . . . ,0, . . .) for n ∈NandT ((kin−1−times︷ ︸︸ ︷0, . . . ,0, i/kn,0, . . .)) = (ki+1n −1−times︷ ︸︸ ︷0, . . . ,0, (i + 1)/kn,0, . . .) for i ∈ {1, . . . ,kn − 1},n ∈ N.The operator P : M(S) → M(S) given by the formula Pμ = T∗(μ) is asymptotically stable, but it does not satisfy the Cesàro e-property at 0. Indeed, if we take an arbitrary continuous function f : S → R+ such that f ((0, . . . , 0, . . .)) = 0 and f (x) = 1 for x ∈ S such that ‖x‖∞ ≥ 1/2, we have1knkn∑i=1U i f ((kn−1︷ ︸︸ ︷0, . . . ,0,1/kn,0, . . .)) − 1knkn∑i=1U i f ((0, . . .)) ≥ 1/2.We are in a position to formulate the main result of our paper (Theorem 2.3).Theorem 2.3. Let P be an asymptotically stable Feller operator and let μ∗ be its unique invariant measure. If IntS(suppμ∗) = ∅, then P satisfies the e-property.Its proof involves the following lemma.Lemma 2.4. Let P be an asymptotically stable Feller operator and let μ∗ be its unique invariant measure. Let U be dual to P . If IntS (suppμ∗) = ∅, then, for every f ∈ Cb(S) and any ε > 0, there exists a ball B ⊂ suppμ∗ and N ∈N such that|Un f (x) − Un f (y)| ≤ ε for any x, y ∈ B, n ≥ N. (3)Proof. Fix f ∈ Cb(S) and ε > 0. Let W be an open set such that W ⊂ suppμ∗ . Set Y = W and observe that the subspace Yis a Baire space. SetYn := {x ∈ Y : |Um f (x) − 〈 f ,μ∗〉| ≤ ε/2 for all m ≥ n}and observe that Yn is closed andY =∞⋃n=1Yn.By the Baire category theorem, there exists N ∈ N such that IntY Y N = ∅. Thus there exists a set V ⊂ Y N open in the space Y and consequently a ball B in S such that B ⊂ Y N ⊂ suppμ∗ . Since|Un f (x) − 〈 f ,μ∗〉| ≤ ε/2 for any x ∈ B and n ≥ N,condition (3) is satisfied. 1250 S.C. Hille et al. / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 1247–1251We are ready to prove Theorem 2.3.Proof. Assume, contrary to our claim, that P does not satisfy the e-property. Therefore there exist a function f ∈ Cb(S) and a point x0 ∈ S such thatlim supx→x0lim supn→∞|Un f (x) − Un f (x0)| > 0.Choose ε > 0 such thatlim supx→x0lim supn→∞|Un f (x) − Un f (x0)| ≥ 3ε.Let B := B(z, 2r) be a ball such that condition (3) holds. Since B(z, r) ⊂ suppμ∗ , we have γ := μ∗(B(z, r)) > 0. Choose α ∈ (0, γ ). Since the operator P is asymptotically stable, we havelim infn→∞ Pnμ(B(z, r)) > α for all μ ∈ P(S), (4)by the Alexandrov theorem (see [4]).Let k ≥ 1 be such that 2(1 −α)k| f | < ε. By induction we are going to define two sequences of measures (νx0i )ki=1, (μx0i )ki=1and a sequence of integers (ni)ki=1 in the following way: let n1 ≥ 1 be such thatPn1δx0(B(z, r)) > α. (5)Choose r1 < r such that Pn1δx0 (B(z, r1)) > α and Pn1δx0 (∂ B(z, r1)) = 0 and setνx01 (·) =Pn1δx0(· ∩ B(z, r1))Pn1δx0(B(z, r1))(6)andμx01 (·) =11 − α(Pn1δx0(·) − ανx01 (·)). (7)Assume that we have done it for i = 1, . . . , l, for some l < k. Now let nl+1 be such thatPnl+1μx0l (B(z, r)) > α. (8)Choose rl+1 < r such that Pnl+1μx0l (B(z, rl+1)) > α and Pnl+1μx0l (∂ B(z, rl+1)) = 0 and setνx0l+1(·) =Pnl+1μx0l (· ∩ B(z, rl+1))Pnl+1μx0l (B(z, rl+1))(9)andμx0l+1(·) =11 − α(Pnl+1μx0l (·) − ανx0l+1(·)). (10)We are done. We havePn1+...+nkδx0(·) = αPn2+...+nkνx01 (·) + α(1 − α)Pn3+...+nkνx02 (·) + . . .++ α(1 − α)k−1νx0k (·) + (1 − α)kμx0k (·).By induction, we check that νxi − νx0i → 0 and μxi − μx0i → 0 weakly as d(x, x0) → 0. Indeed, if i = 1, then νx1 − νx01 → 0weakly (as d(x, x0) → 0), by the fact that P is a Feller operator and limd(x,x0)→0 Pn1δx(B(z, r1)) = Pn1δx0 (B(z, r1)), by the Alexandrov theorem due to the fact that Pn1δx0 (∂ B(z, r1)) = 0. On the other hand, the weak convergence μx1 − μx01 → 0as d(x, x0) → 0 follows directly from the definition of μx1. Moreover, observe that for x sufficiently close to x0, we have Pn1δx(B(z, r)) > α and therefore μx1 ∈P(S).Assume now that we have proved that νxi − νx0i → 0 and μxi − μx0i → 0 weakly as d(x, x0) → 0 for i = 1, . . . , l. We show that νxl+1 − νx0l+1 → 0 and μxl+1 − μx0l+1 → 0 weakly as d(x, x0) → 0 too. Analogously, limd(x,x0)→0 Pnl+1μxl (B(z, rl+1)) =Pnl+1μx0l (B(z, rl+1)), by the Alexandrov theorem due to the fact that Pnl+1μx0l (∂ B(z, rl+1)) = 0 and from the definition of νxl+1, we obtain that νxl+1 −νx0l+1 → 0 weakly as d(x, x0) → 0. The weak convergence μxl+1 −μx0ł+1 → 0 as d(x, x0) → 0 follows now directly from the definition of μxl+1 and for x sufficiently close to x0, we have Pnl+1μxl (B(z, r)) > α and therefore μxl+1 ∈P(S). We are done.Observe that for any x sufficiently close to x0 and all n ≥ n1 + . . . + nk , we havePnδx(·) = αPn−n1νx1(·) + α(1 − α)Pn−n1−n2νx2(·) + . . .+ α(1 − α)k−1 Pn−n1−...−nkνx(·) + (1 − α)k Pn−n1−...−nkμx(·),k kS.C. Hille et al. / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 1247–1251 1251where suppνxi ⊂ B(z, r) for all i = 1, . . . , k. Thuslim supn→∞|〈 f , Pnνxi 〉 − 〈 f , Pnνx0i 〉| = lim supn→∞|〈Un f − 〈 f ,μ∗〉, νxi 〉 − 〈Un f − 〈 f ,μ∗〉, νx0i 〉|≤ ε/2 + ε/2 = ε (11)for all i = 1, . . . , k and x sufficiently close to x0. Hence3ε < lim supx→x0lim supn→∞|Un f (x) − Un f (x0)| = lim supx→x0lim supn→∞|〈 f , Pnδx〉 − 〈 f , Pnδx0〉|≤ ε(α + α(1 − α) + . . . α(1 − α)k−1) + 2(1 − α)k| f |≤ ε + ε = 2ε,which is impossible. This completes the proof. AcknowledgementWe thank Klaudiusz Czudek for providing us with Example 2.1.References[1] D. Czapla, A criterion of asymptotic stability for Markov–Feller e-chains on Polish spaces, Ann. Pol. Math. 105 (3) (2012) 267–291.[2] D. Czapla, K. Horbacz, Equicontinuity and stability properties of Markov chains arising from iterated function systems on Polish spaces, Stoch. Anal. Appl. 32 (1) (2014) 1–29.[3] A. Es-Sarhir, M.K. von Renesse, Ergodicity of stochastic curve shortening flow in the plane, SIAM J. Math. Anal. 44 (1) (2012) 224–244.[4] S.N. Ethier, T.G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, New York, 1985.[5] F.Z. Gong, Y. Liu, Ergodicity and asymptotic stability of Feller semigroups on Polish metric spaces, Sci. China Math. 58 (6) (2015) 1235–1250.[6] B. Jamison, Asymptotic behaviour of successive iterates of continuous functions under a Markov operator, J. Math. Anal. Appl. 9 (1964) 203–214.[7] T. Komorowski, S. Peszat, T. Szarek, On ergodicity of some Markov operators, Ann. Probab. 38 (4) (2010) 1401–1443.[8] A. Lasota, T. Szarek, Lower bound technique in the theory of a stochastic differential equations, J. Differ. Equ. 231 (2006) 513–533.[9] J. Li, S. Tu, X. Ye, Mean equicontinuity and mean sensitivity, Ergod. Theory Dyn. Syst. 35 (8) (2015) 2587–2612.[10] S.P. Meyn, R.L. Tweedie, Markov Chains and Stochastic Stability, Cambridge University Press, Cambridge, UK, 2009.[11] Ł. Stettner, Remarks on ergodic conditions for Markov processes on Polish spaces, Bull. Pol. Acad. Sci., Math. 42 (2) (1994) 103–114.[12] T. Szarek, D.T.H. Worm, Ergodic measures of Markov semigroups with the e-property, Ergod. Theory Dyn. Syst. 32 (3) (2012) 1117–1135.[13] S. Wȩdrychowicz, A. Wiśnicki, On some results on the stability of Markov operators, Stud. Math. (2017), https://doi.org/10.4064/sm8584-3-2017, in press.[14] D.T.H. Worm, Semigroups on Spaces of Measures, PhD thesis, Leiden University, Leiden, The Netherlands, 2010.[15] R. Zaharopol, Invariant Probabilities of Transition Functions, Probability and Its Applications, Springer, Cham, Switzerland, 2014.

Hille, S.C.

Szarek, T.

Ziemlanska, M.A.

Leiden University Scholary Publications

arXiv:1710.02352v1  [math.PR]  6 Oct 2017EQUICONTINUOUS FAMILIES OF MARKOV OPERATORS IN VIEWOF ASYMPTOTIC STABILITYSANDER C. HILLE, TOMASZ SZAREK, AND MARIA A. ZIEMLAŃSKAAbstract. Relation between equicontinuity – the so-called e–property and stability ofMarkov operators is studied. In particular, it is shown that any asymptotically stableMarkov operator with an invariant measure such that the interior of its support is non-empty satisfies the e–property.1. IntroductionThis paper is centered around two concepts of equicontinuity for Markov operators definedon probability measures on Polish spaces: the e-property and the Cesàro e-property. Bothappeared as a condition (among others) in the study of ergodicity of Markov operators.In particular they are very useful in proving existence of a unique invariant measure andits asymptotic stability: at whatever probability measure one starts, the iterates under theMarkov operator will weakly converge to the invariant measure. The first concept appearedin [8, 12] while the second was introduced in [14] as a theoretical generalisation of the firstand allowed the author to extend various results by replacing the e-property condition byapparently weaker the Cesàro e-property condition, among others.Interest in equicontinuous families of Markov operators existed already before the intro-duction of the e-property. Jamison [6], working on compact metric state spaces, introducedthe concepts of (dual) Markov operators on the continuous functions that are ‘uniformlystable’ or ‘uniformly stable in mean’ to obtain a kind of asymptotic stability results inthis setting. Meyn and Tweedie [10] introduced the so-called ‘e-chains’ on locally compactHausdorff topological state spaces, for similar purposes. See also [15] for results in a locallycompact metric setting.The above mentioned concepts were used in proving ergodicity for some Markov chains(see [11, 1, 2, 4, 5, 7, 13]).Date: September 19, 2018.2000 Mathematics Subject Classification. 37A30, 60J05.Key words and phrases. Markov operator, asymptotic stability, e-property, Cesàro e-property, equicon-tinuity, tightness.The work of Maja Ziemlańska has been partially supported by a Huygens Fellowship of Leiden Uni-versity. The work of Tomasz Szarek has been supported by the National Science Centre of Poland, grantnumber 2016/21/B/ST1/00033.12 SANDER C. HILLE, TOMASZ SZAREK, AND MARIA A. ZIEMLAŃSKAIt is worth mentioning here that similar concepts appear in the study of mean equicontinousdynamical systems mainly on compact spaces (see for instance [9]). However it must bestressed here that our space of Borel probability measures defined on some Polish space isnon-compact.Studing the e–property the natural question arose whether any asymptotically stableMarkov operator satisfies this property. Proposition 6.4.2 in [10] says that this holdswhen the phase space is compact. In particular, the authors showed that the strongere–chain property is satisfied. Unfortunately, the proof contains a gap and it is quite easyto construct an example showing that some additional assumptions must be then added.On the other hand, striving to repair the gap of the Meyn-Tweedie result mentioned above,we show that any asymptotically stable Markov operator with an invariant measure suchthat the interior of its support is nonempty satisfies the e–property.2. PreliminariesLet (S, d) be a Polish space. By B(x, r) we denote the open ball in (S, d) of radius r,centered at x ∈ S and ∂B(x, r) denotes its boundary. Further E, IntSE denote the closureof E ⊂ S and the interior of E, respectively. By Cb(S) we denote the vector space ofall bounded real-valued continuous functions on S and by Bb(S) all bounded real-valuedBorel measurable functions, both equipped with the supremum norm | · |. By Lb(S) wedenote the subspace of Cb(S) of all bounded Lipschitz functions (for the metric d on S).For f ∈ Lb(S), Lipf denotes the Lipschitz constant of f .By M(S) we denote the family of all finite Borel measures on S and by P(S) the subfamilyof all probability measures in M(S). For µ ∈ M(S), its support is the setsupp µ := {x ∈ S : µ(B(x, r)) > 0 for all r > 0}.An operator P : M(S) → M(S) is called a Markov operator (on S) if it satisfies thefollowing two conditions:(i) (Positive linearity) P (λ1µ1 + λ2µ2) = λ1Pµ1 + λ2Pµ2for λ1, λ2 ≥ 0; µ1, µ2 ∈ M(S);(ii) (Preservation of the norm) Pµ(S) = µ(S) for µ ∈ M(S).A measure µ∗ is called invariant if Pµ∗ = µ∗. A Markov operator P is asymptoticallystable if there exists a unique invariant measure µ∗ ∈ P(S) such that Pnµ → µ∗ weakly asn → ∞ for every µ ∈ P(S).For brevity we shall use the notation:〈f, µ〉 :=∫Sf(x)µ(dx) for f ∈ Bb(S), µ ∈ M(S).EQUICONTINUOUS FAMILIES OF MARKOV OPERATORS 3A Markov operator P is regular if there exists a linear operator U : Bb(S) → Bb(S) suchthat〈f, Pµ〉 = 〈Uf, µ〉 for all f ∈ Bb(S), µ ∈ M(S).The operator U is called the dual operator of P . A regular Markov operator is a Felleroperator if its dual operator U maps Cb(S) into itself. Equivalently, P is Feller if it iscontinuous in the weak topology (cf. [14], Proposition 3.2.2).A Feller operator P satisfies the e–property at z ∈ S if for any f ∈ Lb(S) we have(1) limx→zlim supn→∞|Unf(x)− Unf(z)| = 0,i.e., if the family of iterates {Unf : n ∈ N} is equicontinuous at z ∈ S. We say that aFeller operator satisfies the e–property if it satisfies it at any z ∈ S.D. Worm slightly generalized the e–property introducing the Cesáro e–property (see [14]).Namely, a Feller operator P will satisfy the Cesáro e–property at z ∈ S if for any f ∈ Lb(S)we have(2) limx→zlim supn→∞∣∣∣∣∣1nn∑k=1Ukf(x)−1nn∑k=1Ukf(z)∣∣∣∣∣= 0.Analogously a Feller operator satisfies the Cesáro e–property if it satisfies this property atany z ∈ S.The following simple example shows that Proposition 6.4.2 in [10] fails.Example 2.1. Let S = {1/n : n ≥ 1} ∪ {0} and let T : S → S be given by the followingformula:T (0) = T (1) = 0 and T (1/n) = 1/(n− 1) for n ≥ 2.The operator P : M(S) → M(S) given by the formula Pµ = T∗(µ) (the pushforwardmeasure) is asymptotically stable but it does not satisfy the e–property at 0.Jamison [6] introduced for a Markov operator the property of uniform stability in meanwhen {Unf : n ∈ N} is an equicontinuous family of functions in the space of real-vauledcontinuous function C(S) for every f ∈ C(S). Here C is a compact metric space. Sincethe space of bounded Lipschitz functions is dense for the uniform norm in the space ofbounded uniform continuous functions, this property coincides with the Cesáro e–propertyfor compact metric spaces. Now, if the Markov operator P on the compact metric spaceis asymptotically stable, with the invariant measure µ∗ ∈ M1, then1n∑ni=1 Uif → 〈f, µ∗〉pointwise, for every f ∈ C(S). According to Theorem 2.3 in [6] this implies that P isuniformly stable in mean, i.e., has the Cesáro e–property.Example 2.2. Let (kn)n≥1 be an increasing sequence of prime numbers. SetS := {(kin−1−times︷ ︸︸ ︷0, . . . , 0, i/kn, 0, . . .) ∈ l∞ : i ∈ {0, . . . , kn}, n ∈ N}.4 SANDER C. HILLE, TOMASZ SZAREK, AND MARIA A. ZIEMLAŃSKAThe set S endowed with the l∞-norm ‖ · ‖∞ is a (noncompact) Polish space. Define T :S → S by the formulaT ((0, . . .)) = T ((kknn −1−times︷ ︸︸ ︷0, . . . , 0, 1, 0, . . .)) = (0, . . . , 0, . . .) for n ∈ NandT ((kin−1−times︷ ︸︸ ︷0, . . . , 0, i/kn, 0, . . .)) = (ki+1n −1−times︷ ︸︸ ︷0, . . . , 0, (i+ 1)/kn, 0, . . .) for i ∈ {1, . . . , kn − 1}, n ∈ N.The operator P : M(S) → M(S) given by the formula Pµ = T∗(µ) is asymptoticallystable but it does not satisfy the Cesáro e–property at 0. Indeed, if we take an arbitrarycontinuous function f : S → R+ such that f((0, . . . , 0, . . .)) = 0 and f(x) = 1 for x ∈ Ssuch that ‖x‖∞ ≥ 1/2 we have1knkn∑i=1U if((kn−1︷ ︸︸ ︷0, . . . , 0, 1/kn, 0, . . .))−1knkn∑i=1U if((0, . . .)) ≥ 1/2.We are in a position to formulate the main result of our paper:Theorem 2.3. Let P be an asymptotically stable Feller operator and let µ∗ be its uniqueinvariant measure. If IntS(supp µ∗) 6= ∅, then P satisfies the e–property.Its proof involves the following lemma:Lemma 2.4. Let P be an asymptotically stable Feller operator and let µ∗ be its uniqueinvariant measure. Let U be dual to P . If IntS(supp µ∗) 6= ∅, then for every f ∈ Cb(S)and any ε > 0 there exists a ball B ⊂ supp µ∗ and N ∈ N such that(3) |Unf(x)− Unf(y)| ≤ ε for any x, y ∈ B, n ≥ N.Proof. Fix f ∈ Cb(S) and ε > 0. Let W be an open set such that W ⊂ supp µ∗. SetY = W and observe that the subspace Y is a Baire space. SetYn := {x ∈ Y : |Umf(x)− 〈f, µ∗〉| ≤ ε/2 for all m ≥ n}and observe that Yn is closed andY =∞⋃n=1Yn.By the Baire category theorem there exist N ∈ N such that IntY YN 6= ∅. Thus thereexists a set V ⊂ YN open in the space Y and consequently a ball B in S such thatB ⊂ YN ⊂ suppµ∗. Since|Unf(x)− 〈f, µ∗〉| ≤ ε/2 for any x ∈ B and n ≥ N,condition (3) is satisfied. We are ready to prove Theorem 2.3.EQUICONTINUOUS FAMILIES OF MARKOV OPERATORS 5Proof. Assume, contrary to our claim, that P does not satisfy the e–property. Thereforethere exist a function f ∈ Cb(S) and a point x0 ∈ S such thatlim supx→x0lim supn→∞|Unf(x)− Unf(x0)| > 0.Choose ε > 0 such thatlim supx→x0lim supn→∞|Unf(x)− Unf(x0)| ≥ 3ε.Let B := B(z, 2r) be a ball such that condition (3) holds. Since B(z, r) ⊂ supp µ∗, we haveγ := µ∗(B(z, r)) > 0. Choose α ∈ (0, γ). Since the operator P is asymptotically stable, wehave(4) lim infn→∞P nµ(B(z, r)) > α for all µ ∈ P(S),by the Alexandrov theorem (see [3]).Let k ≥ 1 be such that 2(1−α)k|f | < ε. By induction we are going to define two sequencesof measures (νx0i )ki=1, (µx0i )ki=1 and a sequence of integers (ni)ki=1 in the following way: letn1 ≥ 1 be such that(5) P n1δx0(B(z, r)) > α.Choose r1 < r such that Pn1δx0(B(z, r1)) > α and Pn1δx0(∂B(z, r1)) = 0 and set(6) νx01 (·) =P n1δx0(· ∩B(z, r1))P n1δx0(B(z, r1))and(7) µx01 (·) =11− α(P n1δx0(·)− ανx01 (·)) .Assume that we have done it for i = 1, . . . , l, for some l < k. Now let nl+1 be such that(8) P nl+1µx0l (B(z, r)) > α.Choose rl+1 < r such that Pnl+1µx0l (B(z, rl+1)) > α and Pnl+1µx0l (∂B(z, rl+1)) = 0 and set(9) νx0l+1(·) =P nl+1µx0l (· ∩ B(z, rl+1))P nl+1µx0l (B(z, rl+1))and(10) µx0l+1(·) =11− α(P nl+1µx0l (·)− ανx0l+1(·)).We are done. We haveP n1+...+nkδx0(·) = αPn2+...+nkνx01 (·) + α(1− α)Pn3+...+nkνx02 (·) + . . .++ α(1− α)k−1νx0k (·) + (1− α)kµx0k (·).By induction we check that νxi −νx0i → 0 and µxi −µx0i → 0 weakly as d(x, x0) → 0. Indeed,if i = 1, then νx1 −νx01 → 0 weakly (as d(x, x0) → 0), by the fact that P is a Feller operatorand limd(x,x0)→0 Pn1δx(B(z, r1)) = Pn1δx0(B(z, r1)), by the Alexandrov theorem due to the6 SANDER C. HILLE, TOMASZ SZAREK, AND MARIA A. ZIEMLAŃSKAfact that P n1δx0(∂B(z, r1)) = 0. On the other hand, the weak convergence µx1 − µx01 → 0as d(x, x0) → 0 follows directly from the definition of µx1. Moreover, observe that for xsufficiently close to x0 we have Pn1δx(B(z, r)) > α and therefore µx1 ∈ P(S).Assume now that we have proved that νxi −νx0i → 0 and µxi −µx0i → 0 weakly as d(x, x0) → 0for i = 1, . . . , l. We show that νxl+1 − νx0l+1 → 0 and µxl+1 − µx0l+1 → 0 weakly as d(x, x0) → 0too. Analogously, limd(x,x0)→0 Pnl+1µxl (B(z, rl+1)) = Pnl+1µx0l (B(z, rl+1)), by the Alexan-drov theorem due to the fact that P nl+1µx0l (∂B(z, rl+1)) = 0 and from the definition ofνxl+1 we obtain that νxl+1 − νx0l+1 → 0 weakly as d(x, x0) → 0. The weak convergenceµxl+1 − µx0l+1 → 0 as d(x, x0) → 0 follows now directly from the definition of µxl+1 and for xsufficiently close to x0 we have Pnl+1µxl (B(z, r)) > α and therefore µxl+1 ∈ P(S). We aredone.Observe that for any x sufficiently close to x0 and all n ≥ n1 + . . .+ nk we haveP nδx(·) = αPn−n1νx1 (·) + α(1− α)Pn−n1−n2νx2 (·) + . . .+ α(1− α)k−1P n−n1−...−nkνxk (·) + (1− α)kP n−n1−...−nkµxk(·),where supp νxi ⊂ B(z, r) for all i = 1, . . . , k. Thuslim supn→∞|〈f, P nνxi 〉 − 〈f, Pnνx0i 〉| = lim supn→∞|〈Unf − 〈f, µ∗〉, νxi 〉 − 〈Unf − 〈f, µ∗〉, νx0i 〉|≤ ε/2 + ε/2 = ε(11)for all i = 1, . . . , k and x sufficiently close to x0. Hence3ε < lim supx→x0lim supn→∞|Unf(x)− Unf(x0)| = lim supx→x0lim supn→∞|〈f, P nδx〉 − 〈f, Pnδx0〉|≤ ε(α+ α(1− α) + . . . α(1− α)k−1) + 2(1− α)k|f |≤ ε+ ε = 2ε,which is impossible. This completes the proof. Acknowledgements. We thank Klaudiusz Czudek for providing us with Example 1.References[1] Czapla, D. (2012), A criterion of asymptotic stability for Markov-Feller e-chains on Polish spaces,Ann. Polon. Math. 105, no. 3, 267–291.[2] Czapla, D. and K. Horbacz (2014), Equicontinuity and stability properties of Markov chains arisingfrom iterated function systems on Polish spaces, Stoch. Anal. Appl. 32, no. 1, 1–29.[3] Ethier, S. N. and Kurtz, T. G. (1985). Markov Processes: Characterization and Convergence Wiley,New York.[4] Es-Sarhir, A. and M. K. von Renesse (2012), Ergodicity of stochastic curve shortening flow in theplane, SIAM J. Math. Anal. 44, no. 1, 224–244.[5] Gong, F.Z. and Y. Liu (2015), Ergodicity and asymptotic stability of Feller semigroups on Polishmetric spaces, Sci. China Math. 58, no. 6, 12351250.[6] Jamison, B. (1964), Asymptotic behaviour of succesive iterates of continuous functions under a Markovoperator, J. Math. Anal. Appl. 9, 203–214.EQUICONTINUOUS FAMILIES OF MARKOV OPERATORS 7[7] Komorowski, T., Peszat, S. and T. Szarek (2010), On ergodicity of some Markov operators, Ann.Prob. 38(4), 1401-1443.[8] Lasota, A. and T. Szarek (2006), Lower bound technique in the theory of a stochastic differentialequations, J. Diff. Equ. 231, 513–533.[9] Li, J., Tu, S. and X. Ye (2015), Mean equicontinuity and mean sensitivity, Ergod. Th. & Dynam. Sys.35, no. 8, 2587–2612.[10] Meyn, S. P. and R. L. Tweedie (2009), Markov Chains and Stochastic Stability, Cambridge UniversityPress, Cambridge.[11] Stettner,  L. (1994), Remarks on ergodic conditions for Markov processes on Polish spaces, Bull. PolishAcad. Sci. Math. 42, no. 2, 103114.[12] Szarek, T. and D.T.H. Worm (2012), Ergodic measures of Markov semigroups with the e-property,Ergod. Th. & Dynam. Sys. 32(3), 1117–1135.[13] Wȩdrychowicz, S. and A. Wísnicki (2017), On some results on the stability of Markov operators,Studia Math., to appear.[14] Worm, D.T.H. (2010), Semigroups on spaces of measures, PhD thesis, Leiden University.[15] Zaharopol, R. (2014), Invariant Probabilities of Transition Functions, in the series ‘Probability andIts Applications’, Cham: Springer.Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Nether-lands, (SH,MZ)E-mail address : {shille,m.a.ziemlanska}@math.leidenuniv.nlInstitute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland,(TS)E-mail address : szarek@intertele.pl

Comptes Rendus Mathématique

http://www.numdam.org/item/10.1016/j.crma.2017.10.019.pdf

Equicontinuous Families of Markov Operators in View of Asymptotic
  Stability

Equicontinuous Families of Markov Operators in View of Asymptotic Stability

Abstract

Similar works

Full text

Available Versions

Crossref

Numérisation de Documents Anciens Mathématiques

Leiden University Scholary Publications

Comptes Rendus Mathématique