Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

$W$-entropy formulas and Langevin deformation of flows on Wasserstein space over Riemannian manifolds (1604.02596v2)

Published 9 Apr 2016 in math.PR

Abstract: We introduce Perelman's $W$-entropy and prove the $W$-entropy formula along the geodesic flow on the $L2$-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.

Summary

We haven't generated a summary for this paper yet.