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 $L^p−L^q$ 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 one third derivatives of the Kramers-Fokker-Planck operator.