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The Smoluchowski-Kramers Limit of Stochastic Differential Equations with Arbitrary State-Dependent Friction

Journal article
Authors Scott Hottovy
Austin McDaniel
Giovanni Volpe
Jan Wehr
Published in Communications in Mathematical Physics
Volume 336
Pages 1259-1283
ISSN 00103616
Publication year 2015
Published at
Pages 1259-1283
Language en
Subject categories Physical Sciences, Mathematics

Abstract

© 2014, Springer-Verlag Berlin Heidelberg. 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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http://www.gu.se/english/research/publication/?publicationId=262188
Utskriftsdatum: 2020-02-27