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W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds

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Speaker: 
Xiangdong Li
Affiliation: 
Chinese Academy of Sciences
Date: 
Tue, 2017-09-12 09:30 - 10:30
Location: 
MPIM Lecture Hall

In this talk, we give an overview of some recent works on the study of the W-entropy for the heat equation of the Witten Laplacian on super-Ricci flows and the Langevin deformation on the Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the Ricci flow, we establish the W-entropy formula for the heat equation of the Witten Laplacian on complete Riemannian manifolds with the CD(K, m)-condition and for the heat equation of the time dependent Witten Laplacian on compact manifolds equipped with a (K, m)-super Ricci flow, where $m\in [n, \infty]$ and $K\in \mathbb{R}$.  Furthermore, we prove an analogue of the W-entropy formula for the Wasserstein geodesic flow which corresponds to the optimal transportation problem on Riemannian manifolds. Our result improves a previous result due to Lott and Villani on the displacement convexity of the Boltzmann-Shannon type entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between
above two $W$-entropy formulas,  we introduce the Langevin deformation of geometric flows, which interpolate the geodesic flow and the gradient flows on the Wasserstein space over Riemannian manifolds, and derive the W-entropy formula for the Langevin deformation. Joint work with Songzi Li.

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