Hypoelliptic multiscale Langevin diffusions: Large deviations, invariant measures and small mass asymptotics

Wenqing Hu, Konstantinos Spiliopoulos

Published 2015-06-19Version 1

We study large deviations properties of the second--order hypoelliptic multiscale Langevin equation. We study the homogenization regime and based on an appropriately chosen parametrization of the mass parameter in terms of the parameter that separates the scales we derive the related large deviations principle and prove that it is consistent with the large deviations behavior of its overdamped counterpart. In particular, we prove that the large deviation rate function converges to the large deviations rate function of the first order Langevin equation as the mass parameter tends to zero. To do so, we prove convergence in mean square sense of the invariant measure that corresponds to the hypoelliptic problem to that of the limiting elliptic problem as the mass goes to zero and similarly of the corresponding PDE "cell problems" that appear in the rate functions due to the homogenization effects. We rigorously obtain an expansion of the solution in terms of 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. In particular, we do not restrict attention to potential drifts and we provide explicit information on the dependence of the norms of interest with respect to the mass parameter.