Study of the Kramers-Fokker-Planck quadratic operator with a constant magnetic field

TL;DR

This paper analyzes the Kramers-Fokker-Planck operator with a magnetic field, providing explicit estimates of the associated semigroup's norm over time and as the magnetic parameter grows large.

Contribution

It offers an exact expression for the semigroup norm and detailed estimates in various time regimes for the quadratic Kramers-Fokker-Planck operator with magnetic field.

Findings

Explicit semigroup norm expression derived

Accurate short and long time estimates provided

Uniform-in-time estimates as magnetic field strength increases

Abstract

We study the quadratic Kramers-Fokker-Planck operator with a constant magnetic field and with a quadratic potential. We describe the exact expression of the norm of the semi-group associated to the operator near the equilibrium. At this level, explicit and accurate estimates of this norm are shown in small and long times as well as uniform-in-time estimates when the magnetic parameter $b$ tends to infinity.