Estimates for Invariant Probability Measures of Degenerate SPDEs with Singular and Path-Dependent Drifts

TL;DR

This paper establishes integrability conditions for degenerate infinite-dimensional PDEs with path-dependent drifts to have regular invariant measures, providing new results even for non-degenerate SDEs.

Contribution

It introduces integrability conditions ensuring existence and uniqueness of invariant measures for degenerate SPDEs with singular, path-dependent drifts, extending previous results.

Findings

Existence and uniqueness of invariant measures for degenerate SPDEs.

Entropy and Sobolev estimates for invariant measures.

Applicability to nonlinear functional SPDEs and degenerate SDEs.

Abstract

In terms of a nice reference probability measure, integrability conditions on the path-dependent drift are presented for (infinite-dimensional) degenerate PDEs to have regular positive solutions. To this end, the corresponding stochastic (partial) differential equations are proved to possess the weak existence and uniqueness of solutions, as well as the existence, uniqueness and entropy estimates of invariant probability measures. When the reference measure satisfies the log-Sobolev inequality, Sobolev estimates are derived for the density of invariant probability measures. Some results are new even for non-degenerate SDEs with path-independent drifts. The main results are applied to nonlinear functional SPDEs and degenerate functional SDEs/SPDEs.