An operator splitting scheme for the fractional kinetic Fokker-Planck equation

TL;DR

This paper introduces an operator splitting scheme for the fractional kinetic Fokker-Planck equation, combining exact fractional diffusion solutions with variational transport approximations, and proves its convergence.

Contribution

The paper presents a novel splitting scheme for FKFPE that integrates exact and variational methods, with a convergence proof and a new variational formulation for kinetic transport.

Findings

Scheme converges to a weak solution of FKFPE

Exact solution of fractional diffusion phase via convolution

Variational scheme minimizes an energy functional with optimal transport

Abstract

In this paper, we develop an operator splitting scheme for the fractional kinetic Fokker-Planck equation (FKFPE). The scheme consists of two phases: a fractional diffusion phase and a kinetic transport phase. The first phase is solved exactly using the convolution operator while the second one is solved approximately using a variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. We prove the convergence of the scheme to a weak solution to FKFPE. As a by-product of our analysis, we also establish a variational formulation for a kinetic transport equation that is relevant in the second phase. Finally, we discuss some extensions of our analysis to more complex systems.