Fokker-Planck Dynamics and Entropies for the Normalized Ricci Flow

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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.