Feynman-Kac formula gor general time dependent stochastic parabolic equation on a bounded domain and applications

TL;DR

This paper develops a Feynman-Kac formula for time-inhomogeneous stochastic parabolic PDEs driven by fractional Gaussian noises, enabling analysis of solution regularity and moment bounds in bounded domains.

Contribution

It introduces a novel Feynman-Kac representation for complex stochastic PDEs with time-dependent coefficients and fractional noises, advancing understanding of their regularity and moment behavior.

Findings

Established Feynman-Kac formula for fractional Gaussian driven PDEs

Proved Hölder regularity in space and time variables

Derived bounds for moments of solutions and diffusion processes

Abstract

This paper establishes a Feynman-Kac formula to represent the solution to general time inhomogeneous stochastic parabolic partial differential equations driven by multiplicative fractional Gaussian noises in bounded domain where L_t is a second order uniformly elliptic operator whose coefficients can depend on time and generates a time inhomoegenous Markov process. The idea is to use the Aronson bounds of fundamental solution of the associated heat kernel and the techniques from Malliavin calculus. The newly obtained Feynman-Kac formula is then applied to establish the Holder regularity in the space and time variables. The dependence on time of the coefficients poses serious challenges and new results about the stochastic differential equations are discovered to face the challenge. An amazing application of the Feynman-Kac formula is about the matching upper and lower bounds for all moments of the solution, an critical tool for the intermittency. For the latter result we need first to establish new small ball like bounds for the diffusion associated with the parabolic differential operator in the bounded domain which is of interest on its own.