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Langevin deformation for Rényi entropy on Wasserstein space over Riemannian manifolds

Rong Lei, Songzi Li, Xiang-Dong Li

Abstract

We introduce the Langevin deformation for the Rényi 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 Rényi 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 Rényi 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.

Langevin deformation for Rényi entropy on Wasserstein space over Riemannian manifolds

Abstract

We introduce the Langevin deformation for the Rényi entropy on the -Wasserstein space over or a Riemannian manifold, which interpolates between the porous medium equation and the Benamou-Brenier geodesic flow on the -Wasserstein space and can be regarded as the compressible Euler equations for isentropic gas with damping. We prove the -entropy-information formulae and the the rigidity theorems for the Langevin deformation for the Rényi entropy on the Wasserstein space over complete Riemannian manifolds with non-negative Ricci curvature or CD-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 Rényi entropy as and respectively. Our results are new even in the case of Euclidean spaces and compact or complete Riemannian manifolds with non-negative Ricci curvature.

Paper Structure

This paper contains 13 sections, 26 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Notations
  4. Main results
  5. Variational formulae for Rényi entropy
  6. $W$-entropy-information formula and rigidity theorem
  7. Reference model
  8. Proof of $W$-entropy(-information) formulae
  9. Two corollaries of the $W$-entropy formula
  10. Hamiltonian and Lagrangian along the Langevin deformation
  11. The convergence results
  12. Convergence as $c$ approaches $0$
  13. Convergence as $c$ approaches $+\infty$

Key Result

Theorem 1.1

Li-Li Let $c \in [0, \infty]$ and $M = \mathbb R^n$ or be a compact Riemannian manifold. Let $\mathcal{V}=\int_M V(\rho)d\mu$, where $V(\rho)=\rho\log\rho$ or $V(\rho)=\frac{1}{\gamma-1}\rho^\gamma$ for $\gamma>1$. Given $(\rho_0,\phi_0) \in T\mathcal{P}_2^{\infty}(M,\mu)$ with $(\rho_0, \phi_0)\in

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • ...and 40 more