# Absolute continuity of solutions to reaction-diffusion equations with multiplicative noise

## TL;DR

This paper proves that solutions to certain reaction-diffusion stochastic PDEs with multiplicative noise have absolutely continuous distributions at fixed points, using Malliavin calculus and well-posedness theory.

## Contribution

It establishes absolute continuity of solutions' laws for a class of nonlinear stochastic PDEs driven by multiplicative noise, extending previous results to more general nonlinearities.

## Key findings

- Solutions have absolutely continuous distributions at fixed points.
- The proof employs Malliavin calculus and mild solution theory.
- Results apply to reaction-diffusion equations on bounded domains.

## Abstract

We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on $L^2(G)$, where $G$ is an open bounded domain in $\mathbb{R}^d$ with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function which is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.