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 \(\mathscr {S}^{\infty -1}\) 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 \(\mathfrak {p}\) 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.