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Entropy and Fisher information in non-convex domains: one chain to rule them all

Jean-Baptiste Casteras, Marco Flaim, Léonard Monsaingeon

TL;DR

On non-convex Riemannian domains, the paper proves that the square root of Fisher information is a strong Wasserstein upper gradient for the entropy, enabling a gradient-flow interpretation without relying on displacement convexity. It establishes a novel short-time Fisher information growth bound along the Neumann heat flow that depends on boundary geometry, and provides an exact chain rule for entropy along AC2 curves under natural Fisher-information finiteness. A regularization scheme based on the Neumann heat flow combined with a meticulous $\\varepsilon,\\delta$-limit procedure yields the chain rule and upper-gradient results, avoiding $\\lambda$-convexity arguments. The results extend the metric-space calculus for gradient flows to non-convex domains by intertwining Sturm's gradient estimates with Wasserstein contraction, with potential implications for non-convex Fokker–Planck models.

Abstract

We prove that the (square root) Fisher information functional is a strong Wasserstein upper gradient of the entropy on non-convex Riemannian domains. This fills a gap in the literature by allowing one to completely dispense from $λ$-displacement convexity arguments. Along the way we establish a novel quantitative short-time control of the Fisher information along the Neumann heat flow, and establish an exact chain rule under stronger $AC_2$ assumptions typically satisfied by curves of measures obtained as limits of JKO schemes.

Entropy and Fisher information in non-convex domains: one chain to rule them all

TL;DR

On non-convex Riemannian domains, the paper proves that the square root of Fisher information is a strong Wasserstein upper gradient for the entropy, enabling a gradient-flow interpretation without relying on displacement convexity. It establishes a novel short-time Fisher information growth bound along the Neumann heat flow that depends on boundary geometry, and provides an exact chain rule for entropy along AC2 curves under natural Fisher-information finiteness. A regularization scheme based on the Neumann heat flow combined with a meticulous -limit procedure yields the chain rule and upper-gradient results, avoiding -convexity arguments. The results extend the metric-space calculus for gradient flows to non-convex domains by intertwining Sturm's gradient estimates with Wasserstein contraction, with potential implications for non-convex Fokker–Planck models.

Abstract

We prove that the (square root) Fisher information functional is a strong Wasserstein upper gradient of the entropy on non-convex Riemannian domains. This fills a gap in the literature by allowing one to completely dispense from -displacement convexity arguments. Along the way we establish a novel quantitative short-time control of the Fisher information along the Neumann heat flow, and establish an exact chain rule under stronger assumptions typically satisfied by curves of measures obtained as limits of JKO schemes.

Paper Structure

This paper contains 10 sections, 7 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations and main results
  3. Proofs and discussion
  4. Proof I (via the Wasserstein contraction)
  5. Proof II (via the gradient estimate)
  6. Step 1: regularization
  7. Step 2: chain rule
  8. Step 3: limit $\delta\to 0$.
  9. Step 4: limit $\varepsilon\to 0$
  10. Appendix

Key Result

Theorem 1

Assume that $M$ is convex, i.e. $S=0$ with $\operatorname{I\!I}(x)\geq 0$, and let $\mathop{\mathrm{Ric}}\nolimits(x)\geq -K$. For any $\mu\in \mathcal{P}(M)$, the Neumann heat-flow $\mu_t=P^*_t\mu$ started from $\mu$ satisfies

Figures (1)

  • Figure 1: flow approximation $\mu^n_t=(P_t\rho^n)V$

Theorems & Definitions (14)

  • Theorem 1: Fisher information decay, convex domains
  • Theorem 2: Fisher information decay, non-convex domains
  • Theorem 3: Fisher is a strong upper gradient
  • Theorem 4: exact chain rule under $AC_2$/Fisher conditions
  • proof : Proof of Theorem \ref{['theo:Fisher_decay_convex']}
  • proof : Proof of Theorem \ref{['theo:Fisher_decay']}
  • proof : Proof of Theorem \ref{['theo:fisher_upper_grad']}
  • proof : Proof of Theorem \ref{['theo:chain_rule_G']}
  • Lemma A.1: Benamou-Brenier formula BBlisini2007characterization
  • proof
  • ...and 4 more