Sets of finite perimeter and the Hausdorff-Gauss measure on the Wiener space

TL;DR

This paper extends the concept of finite perimeter sets and Hausdorff measures to the infinite-dimensional Wiener space, establishing an integration by parts formula involving a Hausdorff-Gauss measure on the measure-theoretic boundary.

Contribution

It introduces the measure-theoretic boundary in Wiener space and formulates an integration by parts formula using a Hausdorff-Gauss measure, adapting geometric measure theory to infinite dimensions.

Findings

Defined measure-theoretic boundary in Wiener space

Derived integration by parts formula with Hausdorff-Gauss measure

Extended finite perimeter concepts to infinite-dimensional setting

Abstract

In Euclidean space, the integration by parts formula for a set of finite perimeter is expressed by the integration with respect to a type of surface measure. According to geometric measure theory, this surface measure is realized by the one-codimensional Hausdorff measure restricted on the reduced boundary and/or the measure-theoretic boundary, which may be strictly smaller than the topological boundary. In this paper, we discuss the counterpart of this measure in the abstract Wiener space, which is a typical infinite-dimensional space. We introduce the concept of the measure-theoretic boundary in the Wiener space and provide the integration by parts formula for sets of finite perimeter. The formula is presented in terms of the integration with respect to the one-codimensional Hausdorff-Gauss measure restricted on the measure-theoretic boundary.