We consider a general class of non-gradient hypoelliptic Langevin diffusions
and study two related questions. The first one is large deviations for
hypoelliptic multiscale diffusions. The second one is small mass asymptotics of
the invariant measure corresponding to hypoelliptic Langevin operators and of
related hypoelliptic Poisson equations. The invariant measure corresponding to
the hypoelliptic problem and appropriate hypoelliptic Poisson equations enter
the large deviations rate function due to the multiscale effects. Based on the
small mass asymptotics we derive that the large deviations behavior of the
multiscale hypoelliptic diffusion is consistent with the large deviations
behavior of its overdamped counterpart. Additionally, we rigorously obtain an
asymptotic expansion of the solution to the related density of the invariant
measure and to hypoelliptic Poisson equations with respect to the mass
parameter, characterizing the order of convergence. The proof of convergence of
invariant measures is of independent interest, as it involves an improvement of
the hypocoercivity result for the kinetic Fokker-Planck equation. We do not
restrict attention to gradient drifts and our proof provides explicit
information on the dependence of the bounds of interest in terms of the mass
parameter
