Stochastic very weak solutions to parabolic equations with singular coefficients

Abstract

A class of stochastic parabolic equations with singular potentials is analyzed within the chaos expansion framework, utilizing the Wick product to handle the multiplication of generalized stochastic processes. The analysis combines the chaos expansion method from white noise analysis with the concept of very weak solutions from partial differential equation theory. The stochastic very weak solution to the parabolic evolution problem is defined, and its existence and uniqueness are established. For sufficiently regular potentials and data, we demonstrate the consistency of the stochastic very weak solution with a stochastic weak solution. An illustrative example is provided, potential applications are reviewed, and future challenges are outlined.