Malliavin Calculus for non Gaussian differentiable measures and surface measures in Hilbert spaces

TL;DR

This paper develops a framework for defining surface measures and integration by parts formulas in Hilbert spaces with non-Gaussian measures, applicable to invariant measures of certain stochastic PDEs.

Contribution

It introduces a method to construct surface measures and establish integration by parts formulas for non-Gaussian measures in infinite-dimensional Hilbert spaces, extending existing Gaussian theory.

Findings

Constructed surface measures in Hilbert spaces with non-Gaussian measures.

Proved integration by parts formulas involving surface integrals.

Applied theory to invariant measures of stochastic PDEs like Burgers and reaction-diffusion equations.

Abstract

We construct surface measures in a Hilbert space endowed with a probability measure $\nu$. The theory fits for invariant measures of some stochastic partial differential equations such as Burgers and reaction--diffusion equations. Other examples are weighted Gaussian measures and special product measures $\nu$ of non Gaussian measures; in this case we exhibit a Markov process having $\nu$ as invariant measure. In any case we prove integration by parts formulae on sublevel sets of good functions (including spheres and hyperplanes) that involve surface integrals.