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Generalised Fisher Information in Defective Fokker-Planck Equations

Anton Arnold, Amit Einav, Tobias Wöhrer

TL;DR

This work develops a Generalised Fisher Information framework for linear Fokker-Planck equations with constant diffusion and drifting, extending the Bakry-Émery method to accommodate defective drift. By decomposing the solution into spectrally close and far components and introducing a two-function Fisher information, the authors establish sharp long-time decay rates for a family of $p$-entropies, with explicit polynomial corrections governed by the maximal defect of eigenvalues. The analysis relies on minimal spectral information and yields a robust approach that adapts to nondegenerate diffusion and, via recent extensions, to degenerate cases. The results bridge entropy methods with hypocoercivity-style decompositions, offering a versatile tool for studying convergence to equilibrium in kinetic-type FP equations and related systems.

Abstract

The goal of this work is to introduce and investigate a generalised Fisher Information in the setting of linear Fokker-Planck equations. This functional, which depends on two functions instead of one, exhibits the same decay behaviour as the standard Fisher information, and allows us to investigate different parts of the Fokker-Planck solution via an appropriate decomposition. Focusing almost exclusively on Fokker-Planck equations with constant drift and diffusion matrices, we will use a modification of the well established Bakry-Emery method with this newly defined functional to provide an alternative proof to the sharp long time behaviour of relative entropies of solutions to such equations when the diffusion matrix is positive definite and the drift matrix is defective. This novel approach is different to previous techniques and relies on minimal spectral information on the Fokker-Planck operator, unlike the one presented the authors' previous work, where powerful tools from spectral theory were needed.

Generalised Fisher Information in Defective Fokker-Planck Equations

TL;DR

This work develops a Generalised Fisher Information framework for linear Fokker-Planck equations with constant diffusion and drifting, extending the Bakry-Émery method to accommodate defective drift. By decomposing the solution into spectrally close and far components and introducing a two-function Fisher information, the authors establish sharp long-time decay rates for a family of -entropies, with explicit polynomial corrections governed by the maximal defect of eigenvalues. The analysis relies on minimal spectral information and yields a robust approach that adapts to nondegenerate diffusion and, via recent extensions, to degenerate cases. The results bridge entropy methods with hypocoercivity-style decompositions, offering a versatile tool for studying convergence to equilibrium in kinetic-type FP equations and related systems.

Abstract

The goal of this work is to introduce and investigate a generalised Fisher Information in the setting of linear Fokker-Planck equations. This functional, which depends on two functions instead of one, exhibits the same decay behaviour as the standard Fisher information, and allows us to investigate different parts of the Fokker-Planck solution via an appropriate decomposition. Focusing almost exclusively on Fokker-Planck equations with constant drift and diffusion matrices, we will use a modification of the well established Bakry-Emery method with this newly defined functional to provide an alternative proof to the sharp long time behaviour of relative entropies of solutions to such equations when the diffusion matrix is positive definite and the drift matrix is defective. This novel approach is different to previous techniques and relies on minimal spectral information on the Fokker-Planck operator, unlike the one presented the authors' previous work, where powerful tools from spectral theory were needed.
Paper Structure (20 sections, 24 theorems, 312 equations)

This paper contains 20 sections, 24 theorems, 312 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Results
  4. An Outline of the Proof of our Main Theorem
  5. The Structure of the Work
  6. Properties of the Fokker-Planck Equation
  7. The Generalised Fisher Information
  8. Proof of Main Results
  9. The Interplay Between Fisher Informations and Regularisation Properties of the Fokker-Planck Equation
  10. Interpolation of Generalised Fisher Information
  11. Hyper-, Upper- and Lower-Contractivity
  12. The Convergence to Equilibrium in Fisher Information
  13. Step 1: Decomposition.
  14. Step 2: The Behaviour of the Generalised Fisher Information on the "Spectrally Close" Part of the Decomposition.
  15. Step 3: The Behaviour of the Generalised Fisher Information on the "Spectrally Far" Part of the Decomposition.
  16. ...and 5 more sections

Key Result

Theorem 1.3

Consider the Fokker-Planck equation eq:fokkerplanck with diffusion and drift matrices ${\bf D}$ and ${\bf C}$ which satisfy Conditions item:cond_semipositive-item:cond_no_invariant_subspace_to_kernel and denote by $f_\infty(x)$ the unit mass equilibrium of the equationi.e. $Lf_\infty(x)=0$ and \int_

Theorems & Definitions (58)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 1.6
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 48 more