A Feynman-Kac Approach for the Spatial Derivative of the Solution to the Wick Stochastic Heat Equation Driven by Time Homogeneous White Noise
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.
A FEYNMAN-KAC APPROACH FOR THE SPATIAL DERIVATIVE OF THE SOLUTION TO THE WICK STOCHASTIC HEAT EQUATION DRIVEN BY TIME HOMOGENEOUS WHITE NOISE HYUN-JUNG KIM AND RAMIRO SCOROLLI arXiv:2112.11051v1 [math.PR] 21 Dec 2021 Abstract. We consider the (unique) mild solution u(t, x) of a 1-dimensional stochastic heat equa-…
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