Yet another heat semigroup characterization of BV functions on Riemannian manifolds

TL;DR

This paper characterizes BV functions on compact Riemannian manifolds using the short-time behavior of the heat semigroup, providing a new perspective through probabilistic and analytic methods.

Contribution

It introduces a novel heat semigroup-based characterization of BV functions on Riemannian manifolds, with two independent proof approaches.

Findings

Total variation equals the heat semigroup limit for BV functions.

Probabilistic and analytic methods both effectively characterize BV functions.

The approach extends classical Euclidean results to Riemannian manifolds.

Abstract

This paper provides a characterization of functions of bounded variation (BV) in a compact Riemannian manifold in terms of the short time behavior of the heat semigroup. In particular, the main result proves that the total variation of a function equals the limit characterizing the space BV. The proof is carried out following two fully independent approaches, a probabilistic and an analytic one. Each method presents different advantages.