Gelfand-Shilov and Gevrey smoothing effect of the Cauchy problem for Fokker-Planck equation

In this work, we study the Cauchy problem for Fokker-Planck equation with the external potential. We show that the solution of the Cauchy problem enjoys Gelfand-Shilov S 1 1 ( R 3 ) regularizing property with respect to the velocity variable and Gevrey G 3 2 ( R 3 ) regularizing property with respect to the position variable.