Stochastic parabolic equations with singular potentials

TL;DR

This paper introduces a new approach for analyzing stochastic parabolic equations with singular potentials, combining chaos expansion and very weak solutions to establish existence, uniqueness, and consistency.

Contribution

It develops the concept of stochastic very weak solutions for singular potential equations, proving their existence, uniqueness, and independence from regularization.

Findings

Existence and uniqueness of stochastic very weak solutions.

Independence of solutions from regularization of singular potential.

Consistency with stochastic weak solutions.

Abstract

In this work we consider a class of stochastic parabolic equations with singular space depending potential, random driving force and random initial condition. For the analysis of these equations we combine the chaos expansion method from the white noise analysis and the concept of very weak solutions. For given stochastic parabolic equation we introduce the notion of a stochastic very weak solution, prove the existence and uniqueness of the very weak solution to corresponding stochastic initial value problem and show its independence of a regularization on given singular potential. In addition, the consistency of a stochastic very weak solution with a stochastic weak solution is shown.