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

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.

There are no changes of the mathematical part from the first version; two recent papers [18, 19] were added in the references

References in corpus (2)

• On the splitting method for some complex-valued quasilinear equations
• Reflecting Ornstein-Uhlenbeck processes on pinned path spaces

Cited by in corpus (4)

• Dirichlet spaces on H-convex sets in Wiener space
• Surface measures and integration by parts formula on levels sets induced by functionals of the Brownian motion in ${\mathbb{R}}^n$
• Infinite dimensional reflecting Ornstein-Uhlenbeck stochastic process
• BV Functions and Sets of Finite Perimeter on Configuration Spaces