  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.
