Nonlinear Feynman-Kac formulae for SPDEs with space-time noise

TL;DR

This paper develops nonlinear Feynman-Kac formulae for semilinear SPDEs with space-time noise using backward doubly stochastic differential equations, enabling probabilistic analysis of solutions including periodic and stationary states.

Contribution

It introduces a novel probabilistic representation for SPDEs with space-time noise via BDSDEs, extending Feynman-Kac formulae to this class of equations.

Findings

Probabilistic interpretation of certain SPDEs with space-time noise.

Construction of random periodic and stationary solutions.

Extension of Feynman-Kac formulae to nonlinear SPDEs.

Abstract

We study a class of backward doubly stochastic differential equations (BDSDEs) involving martingales with spatial parameters, and show that they provide probabilistic interpretations (Feynman-Kac formulae) for certain semilinear stochastic partial differential equations (SPDEs) with space-time noise. As an application of the Feynman-Kac formulae, random periodic solutions and stationary solutions to certain SPDEs are obtained.