Feynman-Kac formula for the heat equation driven by time-homogeneous white noise potential
Probability
2021-08-30 v1
Abstract
We present a Feynman-Kac formula for the -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.
Keywords
Cite
@article{arxiv.2108.12406,
title = {Feynman-Kac formula for the heat equation driven by time-homogeneous white noise potential},
author = {Ramiro Scorolli},
journal= {arXiv preprint arXiv:2108.12406},
year = {2021}
}