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

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 ${1\over p}+{1\over q}=1$, $1< p, q<\infty$. 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 $q\in (1,\infty)$.