Nash-Aronson Estimate for the Linear Kinetic Fokker-Planck equation

We prove a Nash-Aronson-type upper bound on the fundamental solution of the linear kinetic Fokker Planck equation with friction term, distinguishing two regimes. For long times, we derive a Gaussian upper bound matching the classical parabolic estimate, which reflects the averaging of the velocity variable that occurs in this regime. For short times, the fundamental solution is governed by that of the constant-coefficient Kolmogorov equation, with the friction and potential terms negligible.