Existence and regularity of the density for the solution to semilinear dissipative parabolic SPDEs

TL;DR

This paper establishes the existence and smoothness of the probability density for solutions to a nonlinear stochastic heat equation with additive noise, using Malliavin calculus and maximal monotone operator techniques.

Contribution

It provides the first proof of density existence and regularity for solutions to semilinear dissipative parabolic SPDEs with monotone nonlinearities.

Findings

Density exists and is smooth for solutions at fixed points in time and space.

The approach combines Malliavin calculus with maximal monotone operator theory.

Results apply to equations driven by additive Wiener noise with polynomial growth nonlinearities.

Abstract

We prove existence and smoothness of the density of the solution to a nonlinear stochastic heat equation on $L^2(\mathcal{O})$ (evaluated at fixed points in time and space), where $\mathcal{O}$ is an open bounded domain in $\mathbb{R}^d$. The equation is driven by an additive Wiener noise and the nonlinear drift term is the superposition operator associated to a real function which is assumed to be (maximal) monotone, continuously differentiable, and growing not faster than a polynomial. The proof uses tools of the Malliavin calculus combined with methods coming from the theory of maximal monotone operators.