Feynman-Kac formula for heat equation driven by fractional white noise

TL;DR

This paper extends the Feynman-Kac formula to the multidimensional stochastic heat equation driven by fractional Brownian sheets, providing new insights into solution regularity and chaos expansion.

Contribution

It introduces a Feynman-Kac formula for the stochastic heat equation with fractional noise and analyzes the solution's properties using Malliavin calculus.

Findings

Established a Feynman-Kac representation for fractional noise-driven heat equation

Proved the solution's density is smooth and has Hölder regularity

Derived Wiener chaos expansion for the solution

Abstract

We establish a version of the Feynman-Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. We use the techniques of Malliavin calculus to prove that the process defined by the Feynman-Kac formula is a weak solution of the stochastic heat equation. From the Feynman-Kac formula, we establish the smoothness of the density of the solution and the H\"{o}lder regularity in the space and time variables. We also derive a Feynman-Kac formula for the stochastic heat equation in the Skorokhod sense and we obtain the Wiener chaos expansion of the solution.