An integral inequality for the invariant measure of a stochastic reaction--diffusion equation

TL;DR

This paper establishes an integral inequality for the invariant measure of a stochastic reaction-diffusion equation, extending Malliavin Calculus tools and proving the existence of surface measures in infinite-dimensional spaces.

Contribution

It introduces a new integral inequality for the invariant measure and extends Malliavin Calculus identities to this setting, also proving surface measure existence in infinite dimensions.

Findings

Proved an integral inequality for the invariant measure.

Extended Malliavin Calculus integration by parts formula.

Established existence of surface measures for specific sets in infinite-dimensional space.

Abstract

We consider a reaction--diffusion equation perturbed by noise (not necessarily white). We prove an integral inequality for the invariant measure $\nu$ of a stochastic reaction--diffusion equation. Then we discuss some consequences as an integration by parts formula which extends to $\nu$ a basic identity of the Malliavin Calculus. Finally, we prove the existence of a surface measure for a ball and a half-space of $H$.