The aim of this article is to construct solutions to second order in time stochastic partial differential equations and to show hypocoercivity of the corresponding transition semigroups. More generally, we analyze non-linear infinite-dimensional degenerate stochastic differential equations in terms of their infinitesimal generators. In the first part of this article we use resolvent methods developed by Beznea, Boboc and Röckner to construct diffusion processes with infinite lifetime and explicit invariant measures. The processes provide weak solutions to infinite-dimensional Langevin dynamics. The second part deals with a general abstract Hilbert space hypocoercivity method, developed by Dolbeaut, Mouhot and Schmeiser. In order to treat stochastic (partial) differential equations, Grothaus and Stilgenbauer translated these concepts to the Kolmogorov backwards setting taking domain issues into account. To apply these concepts in the context of infinite-dimensional Langevin dynamics we use an essential m-dissipativity result for infinite-dimensional Ornstein–Uhlenbeck operators, perturbed by the gradient of a potential. We allow unbounded diffusion operators as coefficients and apply corresponding regularity estimates. Furthermore, essential m-dissipativity of a non-sectorial Kolmogorov backward operator associated to the dynamic is needed. Poincaré inequalities for measures with densities w.r.t. infinite-dimensional non-degenerate Gaussian measures are studied. Deriving a stochastic representation of the semigroup generated by the Kolmogorov backward operator as the transition semigroup of a diffusion process enables us to show an \(L^2\)-exponential ergodicity result for the weak solution. Finally, we apply our results to explicit infinite-dimensional degenerate diffusion equations.

本文的目的是构造二阶时间随机偏微分方程的解，并显示相应过渡半群的弱矫顽力。更一般地，我们根据无穷小生成元分析非线性无限维退化随机微分方程。在本文的第一部分中，我们使用 Beznea、Boboc 和 Röckner 开发的分解方法来构建具有无限寿命和显式不变测度的扩散过程。这些过程为无限维 Langevin 动力学提供了弱解。第二部分涉及由 Dolbeaut、Mouhot 和 Schmeiser 开发的一般抽象 Hilbert 空间低矫顽力方法。为了处理随机（偏）微分方程，Grothaus 和 Stilgenbauer 将这些概念转化为考虑域问题的 Kolmogorov 向后设置。为了在无限维 Langevin 动力学的背景下应用这些概念，我们对无限维 Ornstein-Uhlenbeck 算子使用基本的 m 耗散结果，受势梯度的扰动。我们允许无界扩散算子作为系数并应用相应的规律性估计。此外，需要与动态关联的非扇形 Kolmogorov 后向算子的基本 m 耗散性。研究了具有无限维非退化高斯测度的密度测度的庞加莱不等式。\(L^2\) -弱解的指数遍历性结果。最后，我们将我们的结果应用于显式无限维退化扩散方程。