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arXiv · 2108.12406

Feynman-Kac formula for the heat equation driven by time-homogeneous white noise potential

Ramiro Scorolli
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Abstract

We present a Feynman-Kac formula for the $1$-dimensional stochastic heat equation (SHE) driven by a time-homogeneous Gaussian white noise potential, where the noise is interpreted in the Wick-It\^o-Skorokhod sense. Our approach consists in constructing a Wong-Zakai-type approximation for the SHE from which we are able to obtain an "approximating Feynman-Kac" representation via the reduction of the approximated SHE to a deterministic partial differential equation (PDE). Then we will show that those "approximating Feynman-Kac" converge to a well defined object we will call "formal Feynman-Kac" representation which happens to coincide with the unique solution of SHE.

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Ramiro Scorolli. 2021-08-27. Feynman-Kac formula for the heat equation driven by time-homogeneous white noise potential. https://arxiv.org/abs/2108.12406

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