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Sharp Decay of the Fisher Information for Degenerate Fokker-Planck Equations

Anton Arnold, Amit Einav, Tobias Wöhrer

TL;DR

We address the sharp decay to equilibrium for degenerate Fokker-Planck equations under the quadratic Fisher information. The method relies on a Hermite-space spectral decomposition that reveals an invariant subspace structure V_m, and a precise link between the generator L and the drift matrix C, enabling a bound on the evolution of the 2-Fisher information I_2^{I} that scales with m and the spectral gap μ of C. The main result shows I_2^{I}(f(t)|f_inf) ≤ C_m (1+t)^{2 n m} e^{-2 m μ t} I_2^{I}(f_0|f_inf) for initial data orthogonal to the first m-1 Hermite subspaces, and that this rate is sharp via data supported in V_m. This provides a precise, sharp quantitative description of the large-time behavior for degenerate diffusion in terms of the spectrum of C and the degeneracy pattern of D, extending nondegenerate results to the degenerate setting.

Abstract

The goal of this work is to find the sharp rate of convergence to equilibrium under the quadratic Fisher information functional for solutions to Fokker-Planck equations governed by a constant drift term and a constant, yet possibly degenerate, diffusion matrix. A key ingredient in our investigation is a recent work of Arnold, Signorello, and Schmeiser, where the $L^2$-propagator norm of such Fokker-Planck equations was shown to be identical to the propagator norm of a finite dimensional ODE which is determined by matrices that are intimately connected to those appearing in the associated Fokker-Planck equations.

Sharp Decay of the Fisher Information for Degenerate Fokker-Planck Equations

TL;DR

We address the sharp decay to equilibrium for degenerate Fokker-Planck equations under the quadratic Fisher information. The method relies on a Hermite-space spectral decomposition that reveals an invariant subspace structure V_m, and a precise link between the generator L and the drift matrix C, enabling a bound on the evolution of the 2-Fisher information I_2^{I} that scales with m and the spectral gap μ of C. The main result shows I_2^{I}(f(t)|f_inf) ≤ C_m (1+t)^{2 n m} e^{-2 m μ t} I_2^{I}(f_0|f_inf) for initial data orthogonal to the first m-1 Hermite subspaces, and that this rate is sharp via data supported in V_m. This provides a precise, sharp quantitative description of the large-time behavior for degenerate diffusion in terms of the spectrum of C and the degeneracy pattern of D, extending nondegenerate results to the degenerate setting.

Abstract

The goal of this work is to find the sharp rate of convergence to equilibrium under the quadratic Fisher information functional for solutions to Fokker-Planck equations governed by a constant drift term and a constant, yet possibly degenerate, diffusion matrix. A key ingredient in our investigation is a recent work of Arnold, Signorello, and Schmeiser, where the -propagator norm of such Fokker-Planck equations was shown to be identical to the propagator norm of a finite dimensional ODE which is determined by matrices that are intimately connected to those appearing in the associated Fokker-Planck equations.
Paper Structure (4 sections, 7 theorems, 44 equations)

This paper contains 4 sections, 7 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Background and the setting of the problem
  3. Main result
  4. Sharp decay of the $2-$Fisher information

Key Result

Theorem 1.3

Assume that the drift and diffusion matrices ${\bf C}$ and ${\bf D}$ satisfy conditions item:cond_semipositive-item:cond_no_invariant_subspace_to_kernel. Then

Theorems & Definitions (15)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • ...and 5 more