Estimates for invariant probability measures of degenerate SPDEs with singular and path-dependent drifts

• Published in: Probability theory and related fields Vol. 172; no. 3-4; pp. 1181 - 1214
• Main Author: Wang, Feng-Yu
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2018; Springer Nature B.V
• Subjects: Drift; Economics; Entropy; Estimates; Finance; Insurance; Integral calculus; Invariants; Management; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Partial differential equations; Probability; Probability Theory and Stochastic Processes; Quantitative Finance; Statistics for Business; Theoretical; Uniqueness; 60J75; 47G20; Sobolev space; 60G52; Functional SDEs; Invariant probability measure; Integrability condition; Density
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-017-0827-4

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.