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

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) Hölder regularity especially in space.