The Kramers-Fokker-Planck equation with a decaying potential in $\mathbb R^n$, $n \ge 4$

TL;DR

This paper analyzes the spectral properties and large-time behavior of solutions to the Kramers-Fokker-Planck equation with decaying potentials in higher dimensions, using microlocal analysis and quantum scattering techniques.

Contribution

It extends spectral and decay results for the Kramers-Fokker-Planck operator to dimensions n ≥ 4, including optimal decay estimates and large-time expansions.

Findings

Established optimal time-decay estimates for n ≥ 5 odd.

Derived large-time solution expansions involving Maxwell-Boltzmann distribution.

Completed previous results for dimensions 1 and 3, leaving open questions for n=2.

Abstract

We use methods from microlocal analysis and quantum scattering to study spectral properties near the threshold zero of the Kramers-Fokker-Planck operator with a decaying potential in $\mathbb R^n$, $n \ge 4$, and deduce the large-time behavior of solutions to the kinetic Kramers-Fokker-Planck equation. For short-range potentials, we establish an optimal time-decay estimate in weighted $L^2$-spaces when $ n\ge 5$ is odd. For potentials decaying like $O(|x|^{-\rho})$ for some $\rho > n-1$, we obtain, for all dimensions $n \ge 4$, a large-time expansion of the solution with the leading term given by the Maxwell-Boltzmann distribution multiplied by the factor $(4\pi t)^{-\frac n 2}$ corresponding to the decay for the heat equation. These results complete those obtained in [16, 22] for dimensions $n=1$ and $3$. The same questions for $n=2$ are still open.