摘要
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→O_(+) is related to the sub-ellipticity with loss of one third derivatives of the Kramers-Fokker-Planck operator.
