Langevin deformation for R\'enyi entropy on Wasserstein space over Riemannian manifolds

We introduce the Langevin deformation for the R\'enyi entropy on the $L^2$-Wasserstein space over $\mathbb{R}^n$ or a Riemannian manifold, which interpolates between the porous medium equation and the Benamou-Brenier geodesic flow on the $L^2$-Wasserstein space and can be regarded as the compressible Euler equations for isentropic gas with damping. We prove the $W$-entropy-information formulae and the the rigidity theorems for the Langevin deformation for the R\'enyi entropy on the Wasserstein space over complete Riemannian manifolds with non-negative Ricci curvature or CD$(0, m)$-condition. Moreover, we prove the monotonicity of the Hamiltonian and the convexity of the Lagrangian along the Langevin deformation of flows. Finally, we prove the convergence of the Langevin deformation for the R\'enyi entropy as $c\rightarrow 0$ and $c\rightarrow \infty$ respectively. Our results are new even in the case of Euclidean spaces and compact or complete Riemannian manifolds with non-negative Ricci curvature.