$W$-entropy formulas and Langevin deformation on the $L^q$-Wasserstein space over Riemannian manifolds

Abstract

We first prove the $W$-entropy formula and rigidity theorem for the geodesic flow on the $L^q$-Wasserstein space over a complete Riemannian manifold with bounded geometry condition. Then we introduce the Langevin deformation on the $L^q$-Wasserstein space over a complete Riemannian manifold, which interpolates between the $p$-Laplacian heat equation and the geodesic flow on the $L^q$-Wasserstein space, where $\frac{1}{p}+\frac{1}{q}=1$, $1<p,q<\infty$. The local existence, uniqueness and regularity of the Langevin deformation on the $L^q$-Wasserstein space over the Euclidean space and a compact Riemannian manifold are proved for $q\in [2,\infty )$. We further prove the $W$-entropy-information formula and the rigidity theorem for the Langevin deformation on the $L^q$-Wasserstein space over an $n$-dimensional complete Riemannian manifold with non-negative Ricci curvature, where $q\in (1,\infty )$.

Remove the statements of results on weighted manifolds, revise and improve other parts