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A Feynman-Kac Approach for the Spatial Derivative of the Solution to the Wick Stochastic Heat Equation Driven by Time Homogeneous White Noise
arxiv.org · saved by 1 readers
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.
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