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

Beznea L, Cimpean I, Röckner M. A new approach to the existence of invariant measures for Markovian semigroups. ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES. 2019;55(2):977-1000.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