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Entropy-minimizing dynamical transport on Riemannian manifolds

Gabriele Bocchi, Alessio Porretta

Abstract

Given a smooth Riemannian manifold $(M,g)$, compact and without boundary, we analyze the dynamical optimal mass transport problem where the cost is given by the sum of the kinetic energy and the relative entropy with respect to a reference volume measure $e^{-V}dx$. Under the only assumption that the prescribed marginals lie in $L^1(M)$, and a lower bound on the Ricci curvature, we characterize the minimal curves as unique weak solutions of the optimality system coupling the continuity equation with a backward Hamilton-Jacobi equation (with source given by $\log (m)$). We give evidence that the entropic cost enhances diffusive effects in the evolution of the optimal densities, proving $L^1\to L^\infty$ regularization in time for any initial-terminal data, and smoothness of the solutions whenever the marginals are positive and smooth. We use displacement convexity arguments (in the Eulerian approach) and gradient bounds from quasilinear elliptic equations. We also prove the convergence of optimal curves towards the classical Wasserstein geodesics, as the entropic term is multiplied by a vanishing parameter, showing that this kind of functionals can be used to build a smoothing approximation of the standard optimal transport problem.

Entropy-minimizing dynamical transport on Riemannian manifolds

Abstract

Given a smooth Riemannian manifold , compact and without boundary, we analyze the dynamical optimal mass transport problem where the cost is given by the sum of the kinetic energy and the relative entropy with respect to a reference volume measure . Under the only assumption that the prescribed marginals lie in , and a lower bound on the Ricci curvature, we characterize the minimal curves as unique weak solutions of the optimality system coupling the continuity equation with a backward Hamilton-Jacobi equation (with source given by ). We give evidence that the entropic cost enhances diffusive effects in the evolution of the optimal densities, proving regularization in time for any initial-terminal data, and smoothness of the solutions whenever the marginals are positive and smooth. We use displacement convexity arguments (in the Eulerian approach) and gradient bounds from quasilinear elliptic equations. We also prove the convergence of optimal curves towards the classical Wasserstein geodesics, as the entropic term is multiplied by a vanishing parameter, showing that this kind of functionals can be used to build a smoothing approximation of the standard optimal transport problem.
Paper Structure (14 sections, 26 theorems, 256 equations)

This paper contains 14 sections, 26 theorems, 256 equations.

Table of Contents

  1. Introduction
  2. Notations and setting of the problem
  3. Optimal transport functional
  4. The optimality system
  5. Displacement convexity estimates
  6. The optimality system as an elliptic equation
  7. Gradient bounds for smooth solutions
  8. Global Lipschitz bound on $u$
  9. Local bound on the density
  10. Existence of smooth solutions for a penalized problem
  11. Existence and regularity of optimal curves
  12. Convergence to Wasserstein geodesic
  13. Appendix A: existence of smooth solutions
  14. Appendix B: proof of Lemma \ref{['giuseppe-villani']}.

Key Result

Theorem 1.1

Let $(M, g)$ be a smooth compact Riemannian manifold without boundary, with $Ric_g(M)$ bounded below. Let $m_0, m_1 \in L^1( M )\cap\mathcal{P}( M )$, and assume that $V\in W^{2,\infty}( M )$, $\varepsilon >0$. Then the functional $\mathcal{F}_\varepsilon$ in func admits a unique minimum, given by $

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 43 more