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arXiv · 1404.2330

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

Scott Hottovy·Austin McDaniel·Giovanni Volpe·Jan Wehr
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Abstract

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 that 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.

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Scott Hottovy, Austin McDaniel, Giovanni Volpe, Jan Wehr. 2014-10-24. The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction. https://doi.org/10.1007/s00220-014-2233-4

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