Long-time behaviour of non-local in time Fokker-Planck equations via the entropy method

TL;DR

This paper studies the long-term behavior of non-local in time Fokker-Planck equations, demonstrating convergence to equilibrium and deriving decay rates using the entropy method, with implications for fractional and ultraslow diffusion models.

Contribution

It extends the entropy method to non-local in time Fokker-Planck equations, providing decay estimates and analyzing the influence of initial data integrability on decay rates.

Findings

Solutions converge in L^1 to the steady state as t→∞.

Decay rates depend on the initial data's integrability.

Results are applicable to time-fractional and ultraslow diffusion cases.

Abstract

We consider a rather general class of non-local in time Fokker-Planck equations and show by means of the entropy method that as $t\to \infty$ the solution converges in $L^1$ to the unique steady state. Important special cases are the time-fractional and ultraslow diffusion case. We also prove estimates for the rate of decay. In contrast to the classical (local) case, where the usual time derivative appears in the Fokker-Planck equation, the obtained decay rate depends on the entropy, which is related to the integrability of the initial datum. It seems that higher integrability of the initial datum leads to better decay rates and that the optimal decay rate is reached, as we show, when the initial datum belongs to a certain weighted $L^2$ space. We also show how our estimates can be adapted to the discrete-time case thereby improving known decay rates from the literature.