Generalised Fisher Information in Defective Fokker-Planck Equations

TL;DR

This paper introduces a generalized Fisher Information for linear Fokker-Planck equations, enabling new analysis of solution decay and long-term behavior with minimal spectral assumptions.

Contribution

It proposes a novel generalized Fisher Information functional and applies a modified Bakry-Emery method to analyze Fokker-Planck equations with defective drift matrices.

Findings

Demonstrates decay behavior similar to standard Fisher information

Provides an alternative proof of long-time solution behavior

Requires minimal spectral information for analysis

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.