BV Functions, Caccioppoli Sets and Divergence Theorem over Wiener Spaces

Ben Goldys and Xicheng Zhang

Abstract

Using finite dimensional approximation, we give a version of the definition of BV functions on abstract Wiener space introduced by Fukushima and Hino. Then, we study Caccioppoli sets in the classical Wiener space and pinned Wiener space, and provide concrete examples of Caccioppoli sets, such as the balls and the level sets of solutions to SDEs. Moreover, without assuming the ray Hamza conditions, we prove the infinite dimensional divergence theorem in any Caccioppoli set for any bounded continuous and $\mathbb{H}$-Lipschitz continuous vector field in the classical Wiener space. In particular, the isoperimetric inequality holds true for Caccioppoli sets.

Keywords: BV function; Caccioppoli set; Divergence theorem; Pinned Wiener space; Isoperimetric inequality.

AMS Subject Classification: Primary 28A75,28C20,26B15,46G12,60H07,60H10.