In this work we investigate the long-time behavior for Markov processes obtained as the unique mild solution to stochastic partial differential equations in a Hilbert space. We analyze the existence and characterization of invariant measures as well as convergence of transition probabilities. While in the existing literature typically uniqueness of invariant measures is studied, we focus on the case where the uniqueness of invariant measures fails to hold. Namely, introducing a 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 apply our results to Lévy driven Ornstein–Uhlenbeck processes, the Heath–Jarrow–Morton–Musiela equation as well as to stochastic partial differential equations with delay.

在这项工作中，我们研究了马尔可夫过程的长期行为，这是希尔伯特空间中随机偏微分方程的唯一温和解。我们分析了不变测度的存在和特征，以及转移概率的收敛性。虽然在现有文献中通常研究不变性度量的唯一性，但我们关注的是不变性度量的唯一性无法成立的情况。即，引入广义耗散条件结合希尔伯特空间的分解，我们证明了依赖于过程初始状态的多个极限分布的存在，并研究了Wasserstein 2维距离中转移概率的收敛性。最后，我们将结果应用到Lévy驱动的Ornstein-Uhlenbeck过程，Heath-Jarrow-Morton-Musiela方程以及具有延迟的随机偏微分方程。