For Hamiltonian systems with small random perturbations, transition densities have uniform positive lower bounds on energy sublevel sets at the slow equilibration time scale.
Small-time asymptotics for hypoelliptic diffusions
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An inductive procedure is developed to calculate the asymptotic behavior at time zero of a diffusion with polynomial drift and degenerate, additive noise. The procedure gives rise to two different rescalings of the process; namely, a functional law of the iterated logarithm rescaling and a distributional rescaling. The limiting behavior of these rescalings is studied, resulting in two related control problems which are solved in nontrivial examples using methods from geometric control theory. The control information from these problems gives rise to a practical criteria for points to be regular on the boundary of a domain in $\mathbf{R}^n$ for such diffusions.
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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.