On integration by parts formula on open convex sets in Wiener spaces

TL;DR

This paper derives an explicit integration by parts formula for open convex sets in Wiener spaces, expressing the boundary measure in terms of the Minkowski functional, extending finite-dimensional results to infinite-dimensional settings.

Contribution

It provides a new explicit formula for the boundary measure density in Wiener spaces for open convex sets, linking it to the Minkowski functional.

Findings

Explicit density formula in terms of Minkowski functional

Extension of finite-dimensional integration by parts to Wiener spaces

Application to open convex sets in infinite-dimensional analysis

Abstract

In Euclidean space, it is well known that any integration by parts formula for a set of finite perimeter $\Omega$ is expressed by the integration with respect to a measure $P(\Omega,\cdot)$ which is equivalent to the one-codimensional Hausdorff measure restricted to the reduced boundary of $\Omega$. The same result has been proved in an abstract Wiener space, typically an infinite dimensional space, where the surface measure considered is the one-codimensional spherical Hausdorff-Gauss measure $\mathscr S^{\infty-1}$ restricted to the measure-theoretic boundary of $\Omega$. In this paper we consider an open convex set $\Omega$ and we provide an explicit formula for the density of $P(\Omega,\cdot)$ with respect to $\mathscr S^{\infty-1}$. In particular, the density can be written in terms of the Minkowski functional $\p$ of $\Omega$ with respect to an inner point of $\Omega$. As a consequence, we obtain an integration by parts formula for open convex sets in Wiener spaces.