Global-in-Time $L^p−L^q$ Estimates for Solutions of the Kramers-Fokker-Planck Equation

Abstract

In this work, we prove an optimal global-in-time Lp−Lq estimate for solutions to the Kramers-Fokker-Planck equation with short range potential in dimension three. Our result shows that the decay rate as t → +∞ is the same as the heat equation in x-variables and the divergence rate as t → 0+ is related to the sub-ellipticity with loss of 1/3 derivatives of the Kramers-Fokker-Planck operator.