A Feynman-Kac approach for the spatial derivative of the solution to the Wick stochastic heat equation driven by time homogeneous white noise

TL;DR

This paper derives a Feynman-Kac representation for the spatial derivative of the solution to a 1D Wick stochastic heat equation driven by white noise, enabling analysis of its regularity.

Contribution

It provides a novel Feynman-Kac type formula for the spatial derivative of the solution, advancing understanding of its regularity and structure.

Findings

Explicit Feynman-Kac representation for the derivative

Chaos expansion reveals optimal Hölder regularity

Enhanced understanding of solution's spatial regularity

Abstract

We consider the (unique) mild solution $u(t,x)$ of a 1-dimensional stochastic heat equation on $[0,T]\times\mathbb R$ driven by time-homogeneous white noise in the Wick-Skorokhod sense. The main result of this paper is the computation of the spatial derivative of $u(t,x)$, denoted by $\partial_x u(t,x)$, and its representation as a Feynman-Kac type closed form. The chaos expansion of $\partial_x u(t,x)$ makes it possible to find its (optimal) H\"older regularity especially in space.