Nash-Aronson Estimate for the Linear Kinetic Fokker-Planck equation

TL;DR

This paper establishes a Nash-Aronson-type upper bound for the fundamental solution of the linear kinetic Fokker-Planck equation, revealing different behaviors in short and long time regimes.

Contribution

It provides the first known Nash-Aronson estimate for this equation, bridging short-time Kolmogorov behavior and long-time Gaussian bounds.

Findings

Gaussian upper bound matches classical parabolic estimate for long times

Fundamental solution governed by Kolmogorov equation at short times

Distinguishes two regimes based on time scale

Abstract

We prove a Nash-Aronson-type upper bound on the fundamental solution of the linear kinetic Fokker Planck equation with friction term, distinguishing two regimes. For long times, we derive a Gaussian upper bound matching the classical parabolic estimate, which reflects the averaging of the velocity variable that occurs in this regime. For short times, the fundamental solution is governed by that of the constant-coefficient Kolmogorov equation, with the friction and potential terms negligible.