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

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.

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Acknowledgements

The author would like to thank Professor Shige Peng for valuable conversations and the referees for helpful comments and a number of corrections.

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Correspondence to Feng-Yu Wang.

Additional information

Supported in part by NNSFC (11771326, 11431014, 11726627).

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Wang, FY. Estimates for invariant probability measures of degenerate SPDEs with singular and path-dependent drifts. Probab. Theory Relat. Fields 172, 1181–1214 (2018). https://doi.org/10.1007/s00440-017-0827-4

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  • DOI: https://doi.org/10.1007/s00440-017-0827-4

Keywords

  • Integrability condition
  • Functional SDEs
  • Invariant probability measure
  • Density
  • Sobolev space

Mathematics Subject Classification

  • 60J75
  • 47G20
  • 60G52