Fractional diffusion limit for a kinetic Fokker-Planck equation with diffusive boundary conditions in the half-line

TL;DR

This paper studies a kinetic Fokker-Planck model with heavy-tailed velocities and diffusive boundary conditions, showing that the rescaled position converges to a reflected stable process and the space-marginal solves a fractional heat equation.

Contribution

It establishes the fractional diffusion limit for a kinetic Fokker-Planck equation with boundary conditions, linking particle dynamics to fractional PDEs.

Findings

Rescaled position converges to a reflected stable process.

Space-marginal converges to a solution of a fractional heat equation.

Provides a kinetic PDE perspective on fractional diffusion limits.

Abstract

We consider a particle living in $\mathbb{R}_+$, whose velocity is a positive recurrent diffusion with heavy-tailed invariant distribution when the particle lives in $(0,\infty)$. When it hits the boundary $x=0$, the particle restarts with a random strictly positive velocity. We show that the properly rescaled position process converges weakly to a stable process reflected on its infimum. From a P.D.E. point of view, the time-marginals of $(X_t, V_t)_{t\geq0}$ solve a kinetic Fokker-Planck equation on $(0,\infty)\times\mathbb{R}_+ \times \mathbb{R}$ with diffusive boundary conditions. Properly rescaled, the space-marginal converges to the solution of some fractional heat equation on $(0,\infty)\times\mathbb{R}_+$.