The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction

Scott Hottovy, Giovanni Volpe, Jan Wehr

Published 2014-04-08, updated 2014-09-19Version 2

We study a class of systems of stochastic differential equations describing diffusive phenomena. The Smoluchowski-Kramers approximation is used to describe their dynamics in the small mass limit. Our systems have arbitrary state-dependent friction and noise coefficients. We identify the limiting equation {and,} in particular, the additional drift term which appears in the limit is expressed in terms of the solution to a Lyapunov matrix equation. The proof uses a theory of convergence of stochastic integrals developed by Kurtz and Protter. The result is sufficiently general to include systems driven by both white and Ornstein-Uhlenbeck colored noises. We discuss applications of the main theorem to several physical phenomena, including the experimental study of Brownian motion in a diffusion gradient.