Large-Scale Regularity for the Periodic Kinetic Fokker-Planck equation

TL;DR

This paper proves a homogenization result for the kinetic Fokker-Planck equation's fundamental solution, showing convergence to an effective heat equation and establishing large-scale regularity with polynomial approximations.

Contribution

It introduces a homogenization framework for the kinetic Fokker-Planck equation and develops large-scale regularity results using polynomial approximations.

Findings

Fundamental solution converges to an effective heat equation in averaged L^2 sense.

Large-scale solutions can be approximated by heterogeneous polynomials with explicit error bounds.

In a broader regime, these polynomials solve the Fokker-Planck equation.

Abstract

We first prove a homogenization result for the fundamental solution of the linear kinetic Fokker Planck equation. We show that this solution converges, in an averaged $L^2$ sense, to the fundamental solution of an effective heat equation with constant effective diffusivity determined by corrector functions solving associated cell problems on the torus. A key feature of the proof is the necessity of second-order correctors to control the averaging of the velocity variable, and the handling of a non-divergence form error term arising from limited spatial regularity of solutions. Additionally, building on this homogenization result, we establish a large-scale regularity result for solutions of this Fokker Planck equation. More specifically, we show that solutions by heterogeneous polynomials, analogous to Taylor polynomials, with an explicit error on large scale domains. Furthermore, we show that in a larger regime, this approximating polynomial solves this Fokker Planck equation.