Construction of a surface integral under local Malliavin assumption and integration by parts formulae

TL;DR

This paper develops a method to define surface measures and integration-by-parts formulas on level sets in infinite-dimensional Hilbert spaces under local Malliavin assumptions, facilitating analysis of Gaussian processes.

Contribution

It introduces a new approach to construct surface measures and integration-by-parts formulas in infinite-dimensional spaces with local Malliavin conditions, extending stochastic analysis tools.

Findings

Defined surface measure on level sets related to Gaussian measure

Established an integration-by-parts formula in Hilbert spaces

Applied the framework to Gaussian stochastic processes

Abstract

In this paper, we consider convex sets $K_r = \{g \ge r\}$ in an infinite dimensional Hilbert space, where $g$ is suitably related to a reference Gaussian measure $\mu$ in $H$. We first show how to define a surface measure on the level sets $\{g = r\}$ that is related to $\mu$. This allows to introduce an integration-by-parts formula in $H$. This formula can be applied in several important constructions, as for instance the case where $\mu$ is the law of a (Gaussian) stochastic process and $H$ is the space of its trajectories