On the Kramers-Fokker-Planck equation with decreasing potentials in dimension one

TL;DR

This paper investigates the spectral properties and long-time behavior of solutions to the Kramers-Fokker-Planck equation with rapidly decreasing potentials in one dimension, revealing resonance phenomena at zero energy.

Contribution

It provides a detailed analysis of low-energy spectral properties and asymptotic solutions for the Kramers-Fokker-Planck operator with decreasing potentials in one dimension.

Findings

Zero energy is always a resonance for rapidly decreasing potentials.

Derived large time asymptotics of solutions in terms of the Maxwellian.

Characterized the spectral structure near zero energy.

Abstract

For quickly decreasing potentials with one position variable, the first threshold zero is always a resonance of the Kramers-Fokker-Planck operator. In this article we study low-energy spectral properties of the operator and calculate large time asymptotics of solutions in terms of the Maxwellian.