Last Multipliers for Riemannian Geometries, Dirichlet Forms and Markov Diffusion Semigroups

Abstract

We start this study with last multipliers and the Liouville equation for a symmetric and non-degenerate tensor field Z of (0, 2)-type on a given Riemannian geometry (M, g) as a measure of how far away is Z from being divergence-free (and hence \(g^C\)-harmonic) with respect to g. The some topics are studied also for the Riemannian curvature tensor of (M, g) and finally for a general tensor field of (1, k)-type. Several examples are provided, some of them in relationship with Ricci solitons. Inspired by the Riemannian setting, we introduce last multipliers in the abstract framework of Dirichlet forms and symmetric Markov diffusion semigroups. For the last framework, we use the Bakry-Emery carre du champ associated to the infinitesimal generator of the semigroup.