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

2015/08/27 by Beznea, Lucian, Cîmpean, Iulian, Röckner, Michael
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1508.06863

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.

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