BV Functions and Sets of Finite Perimeter on Configuration Spaces

TL;DR

This paper develops the theory of BV functions and sets of finite perimeter on infinite-dimensional configuration spaces with Poisson measures, extending geometric measure theory concepts to this setting.

Contribution

It introduces rigorous definitions of codimensional Poisson measures, establishes their properties, and generalizes De Giorgi's identity and Gauss-Green formula to the configuration space.

Findings

Constructed the $m$-codimensional Poisson measure on configuration spaces.

Proved the equivalence of three definitions of BV functions in this setting.

Expressed perimeter measures via the Poisson measure on the reduced boundary.

Abstract

This paper contributes to foundations of the geometric measure theory in the infinite dimensional setting of the configuration space over the Euclidean space $\mathbb R^n$ equipped with the Poisson measure $\pi$. We first provide a rigorous meaning and construction of the $m$-codimensional Poisson measure -- formally written as "$(\infty-m)$-dimensional Poisson measure" -- on the configuration space. We then show that our construction is consistent with potential analysis by establishing the absolute continuity with respect to Bessel capacities. Secondly, we introduce three different definitions of BV functions based on the variational, relaxation and the semigroup approaches, and prove the equivalence of them. Thirdly, we construct perimeter measures and introduce the notion of the reduced boundary. We then prove that the perimeter measure can be expressed by the $1$-codimensional Poisson measure restricted on the reduced boundary, which is a generalisation of De Giorgi's identity to the configuration space. Finally, we construct the total variation measures for BV functions, and prove the Gau{\ss}--Green formula.