Regularity of the law of solutions to the stochastic heat equation with non-Lipschitz reaction term

TL;DR

This paper establishes the existence of probability densities for solutions to a stochastic heat equation with non-Lipschitz reaction terms, using advanced differentiability analysis in weighted functional spaces.

Contribution

It introduces a novel approach to prove density existence for solutions with non-Lipschitz drifts on unbounded domains.

Findings

Existence of density for solutions to the stochastic heat equation with non-Lipschitz drift.

Development of a differentiability framework in weighted functional spaces.

Application of the methodology to unbounded spatial domains.

Abstract

We prove the existence of density for the solution to the multiplicative semilinear stochastic heat equation on an unbounded spatial domain, with drift term satisfying a half-Lipschitz type condition. The methodology is based on a careful analysis of differentiability for a map defined on weighted functional spaces.