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Noise-Induced Drift in Stochastic Differential Equations with Arbitrary Friction and Diffusion in the Smoluchowski-Kramers Limit

Journal article
Authors Scott Hottovy
Giovanni Volpe
Jan Wehr
Published in Journal of Statistical Physics
Volume 146
Pages 762-773
ISSN 00224715
Publication year 2012
Published at
Pages 762-773
Language en
Keywords Brownian motion, Einstein mobility-diffusion relation, Smoluchowski-Kramers approximation, Stochastic differential equations
Subject categories Physical Sciences


We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e. g. Brownian motion. We study the limit where friction effects dominate the inertia, i. e. where the mass goes to zero (Smoluchowski-Kramers limit). Using the Itô stochastic integral convention, we show that the limiting effective Langevin equations has different drift fields depending on the relation between friction and diffusion. Alternatively, our results can be cast as different interpretations of stochastic integration in the limiting equation, which can be parametrized by α∈ℝ. Interestingly, in addition to the classical Itô (α=0), Stratonovich (α=0. 5) and anti-Itô (α=1) integrals, we show that position-dependent α=α(x), and even stochastic integrals with α∉[0,1] arise. Our findings are supported by numerical simulations. © 2012 Springer Science+Business Media, LLC.

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