On The Rates of Decay to Equilibrium in Degenerate and Defective Fokker-Planck Equations

TL;DR

This paper analyzes the long-term decay rates of solutions to defective Fokker-Planck equations, revealing an exponential decay modulated by a polynomial factor, using spectral theory and hypercontractivity techniques.

Contribution

It introduces a novel approach combining spectral theory and hypercontractivity to establish sharp decay rates for defective Fokker-Planck equations, differing from traditional entropy methods.

Findings

Decay rate is exponential times polynomial in time.

Established sharp asymptotic behavior for a family of entropies.

Applied spectral theory and hypercontractivity to analyze decay.

Abstract

We establish sharp long time asymptotic behaviour for a family of entropies to defective Fokker-Planck equations and show that, much like defective finite dimensional ODEs, their decay rate is an exponential multiplied by a polynomial in time. The novelty of our study lies in the amalgamation of spectral theory and a quantitative non-symmetric hypercontractivity result, as opposed to the usual approach of the entropy method.