For Hamiltonian systems with small random perturbations, transition densities have uniform positive lower bounds on energy sublevel sets at the slow equilibration time scale.
H\"{o}lder regularity for hypoelliptic kinetic equations with rough diffusion coefficients
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abstract
This paper is dedicated to the application of the DeGiorgi-Nash-Moser regularity theory to the kinetic Fokker-Planck equation. This equation is hypoelliptic. It is parabolic only in the velocity variable, while the Liouville transport operator has a mixing effect in the position/velocity phase space. The mixing effect is incorporated in the classical DeGiorgi method via the averaging lemmas. The result can be seen as a H\"{o}lder regularity version of the classical averaging lemmas.
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Quantitative positivity of transition densities for random perturbations of Hamiltonian systems
For Hamiltonian systems with small random perturbations, transition densities have uniform positive lower bounds on energy sublevel sets at the slow equilibration time scale.