Fractional diffusion limit for a fractional Vlasov-Fokker-Planck equation

TL;DR

This paper rigorously derives the macroscopic fractional heat equation as a limit of a fractional Vlasov-Fokker-Planck equation, using advanced analytical methods and establishing solution existence and uniqueness.

Contribution

It introduces a novel fractional Vlasov-Fokker-Planck model and provides a rigorous derivation of its macroscopic limit, including existence and uniqueness results.

Findings

Derivation of fractional heat equation as a macroscopic limit

Establishment of existence and uniqueness of solutions

Application of modified test function method and entropy bounds

Abstract

This paper is devoted to the rigorous derivation of the macroscopic limit of a Vlasov-Fokker-Planck equation in which the Laplacian is replaced by a fractional Laplacian. The evolution of the density is governed by a fractional heat equation with the addition of a convective term coming from the external force. The analysis is performed by a modified test function method and by obtaining a priori estimates from quadratic entropy bounds. In addition, we give the proof of existence and uniqueness of solutions to the Vlasov-fractional-Fokker-Planck equation.