Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A new approach to the existence of invariant measures for Markovian semigroups

Lucian Beznea, Iulian Cîmpean, and Michael Röckner

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Abstract

We give a new, two-step approach to prove existence of finite invariant measures for a given Markovian semigroup. First, we fix a convenient auxiliary measure and then we prove conditions equivalent to the existence of an invariant finite measure which is absolutely continuous with respect to it. As applications, we obtain a unifying generalization of different versions for Harris’ ergodic theorem which provides an answer to an open question of Tweedie. Also, we show that for a nonlinear SPDE on a Gelfand triple, the strict coercivity condition is sufficient to guarantee the existence of a unique invariant probability measure for the associated semigroup, once it satisfies a Harnack type inequality with power. A corollary of the main result shows that any uniformly bounded semigroup on $L^{p}$ possesses an invariant measure and we give some applications to sectorial perturbations of Dirichlet forms.

Résumé

On établit une approche en deux étapes pour démontrer l’existence des mesure invariantes finies pour un semigroupe de Markov donné. En fixant d’abord une mesure auxiliaire convenable, on démontre ensuite des conditions équivalentes à l’existence d’une mesure invariante finie qui est absolument continue par rapport à elle. Comme applications, on obtient une généralisation unificatrice des diverses versions du théorème ergodique de Harris et on fournit une réponse à une question ouverte de Tweedie. On montre aussi que pour une EDP stochastique sur un triplet de Gelfand, la condition de coercivité stricte est suffisante pour garantir l’existence d’une seule mesure de probabilité pour le semigroupe associé, si une inégalité de type Harnack avec puissance est satisfaite. Un corollaire du résultat central montre que tout semigroupe uniformément borné sur $L^{p}$ possède une mesure invariante ; on donne des applications aux perturbations sectorielles des formes de Dirichlet.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 977-1000.

Dates
Received: 14 November 2016
Revised: 27 January 2018
Accepted: 12 April 2018
First available in Project Euclid: 14 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1557820838

Digital Object Identifier
doi:10.1214/18-AIHP905

Subjects
Primary: 37C40: Smooth ergodic theory, invariant measures [See also 37Dxx]
Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35} 37L40: Invariant measures 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J25: Continuous-time Markov processes on general state spaces 31C25: Dirichlet spaces 82B10: Quantum equilibrium statistical mechanics (general)

Keywords
Invariant measure Markovian semigroup Transition function Lyapunov function Krylov–Bogoliubov theorem Harris’ ergodic theorem Uniformly bounded $C_{0}$-semigroup on $L^{p}$ Komlós lemma Sobolev inequality

Citation

Beznea, Lucian; Cîmpean, Iulian; Röckner, Michael. A new approach to the existence of invariant measures for Markovian semigroups. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 977--1000. doi:10.1214/18-AIHP905. https://projecteuclid.org/euclid.aihp/1557820838


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References

  • [1] L. Beznea and N. Boboc. Potential Theory and Right Processes. Mathematics and Its Applications 572. Kluwer, Dordrecht, 2004.
  • [2] L. Beznea and I. Cîmpean. Invariant, super and quasi-martingale functions of a Markov process. In Stochastic Partial Differential Equations and Related Fields 421–434. Springer Proceedings in Mathematics & Statistics 229. Springer, Berlin, 2018.
  • [3] L. Beznea, I. Cîmpean and M. Röckner. Irreducible recurrence, ergodicity, and extremality of invariant measures for resolvents. Stochastic Process. Appl. 128 (2018) 1405–1437.
  • [4] V. Bogachev, M. Röckner and T. S. Zhang. Existence and uniqueness of invariant measures: An approach via sectorial forms. Appl. Math. Optim. 41 (2000) 87–109.
  • [5] J. K. Brooks and R. V. Chacon. Continuity and compactness of measures. Adv. Math. 37 (1980) 16–26.
  • [6] G. Da Prato and J. Zabczyk. Ergodicity for Infinite Dimensional Systems. LMS Lecture Notes 229. Cambridge Univ. Press, Cambridge, 1996.
  • [7] R. Douc, G. Fort and A. Guillin. Subgeometric rates of convergence of $f$-ergodic strong Markov processes. Stochastic Process. Appl. 119 (2009) 897–923.
  • [8] L. C. Florescu. Weak compactness results on $L^{1}$. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 45 (1999) 75–86.
  • [9] S. R. Foguel. Positive operators on $C(X)$. Proc. Amer. Math. Soc. 22 (1969) 295–297.
  • [10] G. Fonseca and R. L. Tweedie. Stationary measures for non-irreducible non-continuous Markov chains with time series applications. Statist. Sinica 12 (2001) 651–660.
  • [11] M. Hairer. Convergence of Markov processes. Lecture notes, Univ. Warwick, 2010. Available at http://www.hairer.org/notes/Convergence.pdf.
  • [12] M. Hairer and J. C. Mattingly. Yet another look at Harris’ ergodic theorem for Markov chains. In Seminar on Stochastic Analysis, Random Fields and Applications VI. Progress in Probability 63 109–117. Springer, Basel, 2011.
  • [13] T. E. Harris. The existence of stationary measures for certain Markov processes. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2: Contributions to Probability Theory 113–124. Univ. California Press, Berkeley, CA, 1956.
  • [14] M. Hino. Existence of invariant measures for diffusion processes on a Wiener space. Osaka J. Math. 35 (1998) 717–734.
  • [15] M. Hino. Exponential decay of positivity preserving semigroups on $L^{p}$. Osaka J. Math. 37 (2000) 603–624.
  • [16] N. Jacob and R. L. Schilling. Towards an $L^{p}$ potential theory for sub-Markovian semigroups: Kernels and capacities. Acta Math. Sin. (Engl. Ser.) 22 (2006) 1227–1250.
  • [17] J. Komlós. A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hung. 18 (1967) 217–229.
  • [18] T. Komorowski, S. Peszat and T. Szarek. On ergodicity of some Markov processes. Ann. Probab. 38 (2010) 1401–1443.
  • [19] N. V. Krylov and B. L. Rozovskii. Stochastic evolution equations. Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. 14 (1979) 71–146.
  • [20] A. Lasota and M. C. Mackey. Probabilistic Properties of Deterministic Systems. Cambridge Univ. Press, Cambridge, 1985.
  • [21] A. Lasota and T. Szarek. Lower bound technique in the theory of a stochastic differential equation. J. Differential Equations 231 (2006) 513–533.
  • [22] W. Liu. Harnack inequality and applications for stochastic evolution equations with monotone drifts. J. Evol. Equ. 9 (2009) 747–770.
  • [23] W. Liu and M. Röckner. SPDE in Hilbert space with locally monotone coefficients. J. Funct. Anal. 259 (2010) 2902–2922.
  • [24] Z. M. Ma and M. Röckner. An Introduction to the Theory of (non-symmetric) Dirichlet Forms. Springer, Berlin, 1992.
  • [25] A. Maitra. Integral representations of invariant measures. Trans. Amer. Math. Soc. 229 (1977) 209–225.
  • [26] S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. Springer, London, 1993.
  • [27] S. P. Meyn and R. L. Tweedie. Stability of Markovian processes II: Continuous-time processes and sampled chains. Adv. in Appl. Probab. 25 (1993) 487–517.
  • [28] S. P. Meyn and R. L. Tweedie. Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes. Adv. in Appl. Probab. 25 (1993) 518–548.
  • [29] S. P. Meyn and R. L. Tweedie. Generalized resolvents and Harris recurrence of Markov processes. In Doeblin and Modern Probability 227–250. Contemp. Math. 149, 1993.
  • [30] C. Prévôt and M. Röckner. A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics 1905. Springer, New York, 2007.
  • [31] M. Röckner and G. Trutnau. A remark on the generator of a right-continuous Markov process. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (2007) 633–640.
  • [32] L. Stettner. On the existence and uniqueness of invariant measure for continuous time Markov processes. Technical report, LCDS, Brown Univ., Providence, RI, 1986.
  • [33] R. L. Tweedie. Drift conditions and invariant measures for Markov chains. Stochastic Process. Appl. 92 (2001) 345–354.
  • [34] I. Vrkoc. A dynamical system in a Hilbert space with a weakly attractive nonstationary point. Math. Bohem. 118 (1993) 401–423.
  • [35] F.-Y. Wang. Functional Inequalities, Markov Semigroups, and Spectral Theory. Science Press, Beijing, 2005.
  • [36] F.-Y. Wang. Harnack Inequalities for Stochastic Partial Differential Equations. Springer, New York, 2013.