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

TL;DR

This paper establishes the precise rate at which solutions to certain degenerate Fokker-Planck equations converge to equilibrium in terms of Fisher information, extending understanding of their long-term behavior.

Contribution

It provides the sharp decay rate of Fisher information for degenerate Fokker-Planck equations with constant drift and diffusion matrices, building on recent operator norm results.

Findings

Derived the exact convergence rate in Fisher information for degenerate Fokker-Planck equations.

Connected the decay rate to the propagator norm of an associated finite-dimensional ODE.

Extended previous results to cases with degenerate diffusion matrices.

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.