On a class of stochastic partial differential equations with multiple invariant measures

Abstract: In this work we investigate the long-time behavior, that is the existence and characterization of invariant measures as well as convergence of transition probabilities, for Markov processes obtained as the unique mild solution to stochastic partial differential equations in a Hilbert space. Contrary to the existing literature where typically uniqueness of invariant measures is studied, we focus on the case where the uniqueness of invariant measures fails to hold. Namely, using a \textit{generalized dissipativity condition} combined with a decomposition of the Hilbert space, we prove the existence of multiple limiting distributions in dependence of the initial state of the process and study the convergence of transition probabilities in the Wasserstein 2-distance. Finally, we show that these results contain L\'evy driven Ornstein-Uhlenbeck processes, the Heath-Jarrow-Morton-Musiela equation as well as stochastic partial differential equations with delay as a particular case.