BV Functions and Sets of Finite Perimeter on Configuration Spaces

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.