论文标题
在一类带有多个不变测量的随机部分微分方程上
On a class of stochastic partial differential equations with multiple invariant measures
论文作者
论文摘要
在这项工作中，我们研究了长期行为，即不变措施的存在和表征以及过渡概率的收敛性，用于作为希尔伯特空间中随机部分微分方程的独特温和解决方案获得的马尔可夫过程。与通常研究不变措施唯一性的现有文献相反，我们专注于不变措施的独特性无法实现的情况。也就是说，使用\ textIt {广义耗散条件}与希尔伯特空间的分解结合在一起，我们证明存在多个限制分布的存在，以依赖于过程的初始状态并研究Wassersteant 2距离中的过渡概率的收敛。最后，我们表明这些结果包含Lévy驱动的Ornstein-Uhlenbeck过程，Heath-Jarrow-Morton-Musiela方程以及随机部分微分方程，并以延迟为特定情况。
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 Lévy driven Ornstein-Uhlenbeck processes, the Heath-Jarrow-Morton-Musiela equation as well as stochastic partial differential equations with delay as a particular case.
