On Initial-Boundary Value Problem of Stochastic Heat Equation in a Lipschitz Cylinder

TL;DR

This paper investigates the regularity of solutions to a stochastic heat equation with Lipschitz boundary conditions, employing advanced mathematical tools to handle heavy random perturbations in the derivative.

Contribution

It establishes new regularity results for stochastic heat equations in Lipschitz cylinders, integrating potential theory, harmonic analysis, and probability methods.

Findings

Proves regularity of solutions under non-zero boundary conditions.

Identifies suitable function spaces for solutions and data.

Utilizes potential theory and harmonic analysis techniques.

Abstract

We consider the initial boundary value problem of non-homogeneous stochastic heat equation. The derivative of the solution with respect to time receives heavy random perturbation. The space boundary is Lipschitz and we impose non-zero cylinder condition. We prove a regularity result after finding suitable spaces for the solution and the pre-assigned datum in the problem. The tools from potential theory, harmonic analysis and probability are used. Some Lemmas are as important as the main Theorem.