Stochastic very weak solution to parabolic equations with singular coefficients

TL;DR

This paper introduces a novel concept of stochastic very weak solutions for parabolic equations with singular coefficients, combining chaos expansion and weak solution theories to establish existence, uniqueness, and consistency.

Contribution

It develops a new framework for analyzing stochastic parabolic equations with singular potentials using chaos expansion and very weak solutions, extending existing methods.

Findings

Existence and uniqueness of stochastic very weak solutions are proven.

Consistency with classical stochastic weak solutions is established for regular potentials.

An illustrative example demonstrates the applicability of the method.

Abstract

A class of stochastic parabolic equations with singular potentials is analysed in the chaos expansion setting where the Wick product is used to give sense to the product of generalized stochastic processes. For the analysis of such equations we combine the chaos expansion method from the white noise analysis and the concept of very weak solutions from the theory of partial differential equations. The stochastic very weak solution to the stochastic parabolic evolution problem is defined and its existence and uniqueness are shown. For regular enough potentials and data we prove consistency of stochastic very weak solution with a stochastic weak solution. We give an example to illustrate the method and possible applications.