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

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 ${\mathcal{S}}^{\infty -1}$ 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 ${\mathcal{S}}^{\infty -1}$. 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.