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 $\rightarrow$ +$\infty$ is the same as the heat equation in x-variables and the divergence rate as t $\rightarrow$ 0 + is related to the sub-ellipticity with loss of 1/3 derivatives of the Kramers-Fokker-Planck operator.