## On the $W$-entropy formula for the Langevin deformation on Wasserstein space

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### Abstract

Inspired by Perelman’s work on the entropy formula for Ricci flow, we introduce the $W$-entropy and prove its monotonicity and rigidity theorem for the geodesics flow on the Wasserstein space over Riemannian manifold. This improves an earlier result due to Lott and Villani on the displacement of the Boltzmann type entropy on Riemannian manifolds with non-negative Ricci curvature. We then introduce the Langevin deformation on the Wasserstein space, which interpolates the Wasserstein geodesic flow and the gradient flow of the Boltzmann entropy (i.e., the heat equation on the underlying manifold). Moreover, we present a $W$-entropy type formula for the Langevin deformation. Joint work with Songzi Li.