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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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.


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

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

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

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Digital Object Identifier

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)

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


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.

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