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

TL;DR

This paper introduces a novel two-step method for establishing the existence of finite invariant measures for Markovian semigroups, with applications to nonlinear SPDEs and sectorial perturbations.

Contribution

It presents a new approach involving auxiliary measures and conditions for invariant measures, generalizing Harris ergodic theorem and addressing open questions.

Findings

Short proof for Lasota and Szarek's invariant measures result

Generalization of Harris ergodic theorem

Existence of invariant measures for nonlinear SPDEs with coercivity and Harnack inequality

Abstract

We give a new, two-step approach to prove existence of finite invariant measures for a given Markovian semigroup. First, we identify 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 give a short proof for the result of Lasota and Szarek on invariant measures and we obtain a unifying generalization of different versions for Harris ergodic theorem which provides an answer to an open question of Tweedie. 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. A corollary of the main result shows that any uniformly bounded semigroup on $L^p$ possesses an invariant measures and we give some applications to sectorial perturbations of Dirichlet forms.