Diffusion approximation for Fokker Planck with heavy tail equilibria : a spectral method in dimension 1

TL;DR

This paper studies the diffusion limit of a 1D Fokker-Planck equation with heavy tail equilibria, showing it converges to a fractional diffusion equation and calculating the diffusion coefficient.

Contribution

It extends previous work by deriving the fractional diffusion limit for 1<beta<5 and explicitly computing the diffusion coefficient.

Findings

The limit diffusion involves a fractional Laplacian with order (beta+1)/6.

The diffusion coefficient kappa is explicitly computed.

The results generalize previous cases for beta>5 and beta=5.

Abstract

This paper is devoted to the diffusion approximation for the 1-d Fokker Planck equation with a heavy tail equilibria of the form (1+v^2)^{-\beta/2}, in the range beta\in ]1,5[. We prove that the limit diffusion equation involves a fractional Laplacian kappa|\Delta|^{\frac{\beta+1}{6}}, and we compute the value of the diffusion coefficient kappa. This extends previous results of E. Nasreddine and M. Puel in the case beta>5, and of P. Cattiaux, E. Nasreddine and M. Puel in the case beta=5.