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 [Formula: see text] of a one-dimensional stochastic heat equation on [Formula: see text] 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 [Formula: see text], denoted by [Formula: see text], and its representation as a Feynman–Kac type closed form. The chaos expansion of [Formula: see text] makes it possible to find its (optimal) Hölder regularity especially in space.