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

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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