On functions of bounded variation on convex domains in Hilbert spaces

TL;DR

This paper explores functions of bounded variation on convex domains within infinite-dimensional Hilbert spaces, linking their total variation to the behavior of a perturbed Ornstein-Uhlenbeck semigroup.

Contribution

It establishes a connection between total variation of BV functions and the short-time behavior of a perturbed Ornstein-Uhlenbeck semigroup in infinite-dimensional Hilbert spaces.

Findings

Total variation relates to the semigroup's short-time behavior.

Provides a new characterization of BV functions in infinite dimensions.

Links geometric measure theory with stochastic semigroup analysis.

Abstract

We study functions of bounded variation (and sets of finite perimeter) on a convex open set $\Omega\subseteq X$, $X$ being an infinite dimensional real Hilbert space. We relate the total variation of such functions, defined through an integration by parts formula, to the short-time behaviour of the semigroup associated with a perturbation of the Ornstein--Uhlenbeck operator.