Fokker-Planck Dynamics and Entropies for the Normalized Ricci Flow
Abstract
Description
We consider some elementary aspects of the geometry of the space of probability measures endowed with Wasserstein distance. In such a setting, we discuss the various terms entering Perelman's shrinker entropy, and characterize two new monotonic functionals for the volume-normalized Ricci flow. One is obtained by a rescaling of the curvature term in the shrinker entropy. The second is associated with a gradient flow obtained by adding a curvature-drift to Perelman's backward heat equation. We show that the resulting Fokker-Planck PDE is the natural diffusion flow for probability measures absolutely continuous with respect to the Ricci-evolved Riemannian measure, we discuss its exponential trend to equilibrium, and its relation with the viscous Hamilton-Jacobi equation.
48 pages, Latex;. This is the final version of the paper. References have been updated. This version contains an extended analysis of the relation between Fokker-Planck dynamics, along the backward Ricci flow, and the Wasserstein geometry on the space of probability measures
48 pages, Latex;. This is the final version of the paper. References have been updated. This version contains an extended analysis of the relation between Fokker-Planck dynamics, along the backward Ricci flow, and the Wasserstein geometry on the space of probability measures