$W$-entropy formulas and Langevin deformation of flows on Wasserstein space over Riemannian manifolds

TL;DR

This paper introduces a $W$-entropy formula for flows on Wasserstein space over Riemannian manifolds, explores Langevin deformation interpolating between gradient and geodesic flows, and establishes related rigidity results.

Contribution

It develops a new $W$-entropy formula for Wasserstein space flows and introduces Langevin deformation, connecting gradient and geodesic flows with applications to Riemannian geometry.

Findings

Proved $W$-entropy formula along Wasserstein geodesic flow.

Established existence, uniqueness, and regularity of Langevin deformation.

Proved convergence and rigidity results for the $W$-entropy.

Abstract

We introduce Perelman's $W$-entropy and prove the $W$-entropy formula along the geodesic flow on the $L^2$-Wasserstein space over compact Riemannian manifolds equipped with Otto's Riemannian metric, which allows us to recapture a previous result due to Lott and Villani on the displacement convexity of $s{\rm Ent}+ns\log s$ on $P^\infty_2(M)$ over Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between the $W$-entropy formula for the geodesic flow on the Wasserstein space and the $W$-entropy formula for the heat equation of the Witten Laplacian on the underlying manifolds, we introduce the Langevin deformation of flows on the Wasserstein space over Riemannian manifold, which interpolates the gradient flow and the geodesic flow on the Wasserstein space over Riemannian manifolds, and can be regarded as the potential flow of the compressible Euler equation with damping on manifolds. We prove the existence, uniqueness and regularity of the Langevin deformation on the Wasserstein space over the Euclidean space and compact Riemannian manifolds, and prove the convergence of the Langevin deformation for $c\rightarrow 0$ and $c\rightarrow \infty$ respectively. We prove an analogue of the Perelman type $W$-entropy formula along the Langevin deformation on the Wasserstein space on Riemannian manifolds. A rigidity theorem is proved for the $W$-entropy for the geodesic flow, and a rigidity model is also provided for the Langevin deformation on the Wasserstein space over complete Riemannian manifolds with the $CD(0, m)$-condition.