Sets of finite perimeter on Riemannian manifolds and stochastic completeness

We prove a heat semigroup characterization of the total variation for compactly supported $\mathrm{B}\mathrm{V}$ on arbitrary smooth complete weighted Riemannian manifolds, extending the main result in \cite{GP15}. We then provide an example of a weighted manifold where such equivalence does not hold for a large class of sets of finite perimeter.