The Kramers-Fokker-Planck equation with a decaying potential in $\mathbb R^n$, $n \ge 4$
Xinghong Pan, Xue Ping Wang, Lu Zhu
TL;DR
This work analyzes the Kramers-Fokker-Planck equation with a decaying potential in ${\mathbb R}^n$, $n\ge4$, using microlocal analysis and quantum scattering to perform threshold spectral analysis near $0$ for the non-selfadjoint operator $P=P_0+W$. It develops low-energy expansions of the free and full resolvents, distinguishes odd and even dimensions, and establishes precise large-time asymptotics: for short-range potentials with $\rho>1$ and $n\ge5$ odd it proves optimal time decay in weighted spaces, while for decays $|x|^{-\rho}$ with $\rho>n-1$ and $n\ge4$ it derives a large-time expansion whose leading term is the Maxwell-Boltzmann projection $\frac{1}{(4\pi t)^{n/2}}\langle \cdot,\mathfrak M\rangle\mathfrak M$. The results yield dispersive $L^p-L^q$ estimates and extend prior work from $n=1,3$ to higher dimensions, underscoring the role of threshold phenomena in non-confining KFP dynamics.
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 $O(|x|^{-ρ})$ for some $ρ> n-1$, 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 $(4πt)^{-\frac n 2}$ 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.