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Analysis of White Noise Limits for Stochastic Systems with Two Fast Relaxation Times

Pavliotis, G. A. and Stuart, A. M. (2005) Analysis of White Noise Limits for Stochastic Systems with Two Fast Relaxation Times. Multiscale Modeling and Simulation, 4 (1). pp. 1-35. ISSN 1540-3459. doi:10.1137/040610507.

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In this paper we present a rigorous asymptotic analysis for stochastic systems with two fast relaxation times. The mathematical model analyzed in this paper consists of a Langevin equation for the particle motion with time-dependent force constructed through an infinite dimensional Gaussian noise process. We study the limit as the particle relaxation time as well as the correlation time of the noise tend to zero, and we obtain the limiting equations under appropriate assumptions on the Gaussian noise. We show that the limiting equation depends on the relative magnitude of the two fast time scales of the system. In particular, we prove that in the case where the two relaxation times converge to zero at the same rate there is a drift correction, in addition to the limiting Itô integral, which is not of Stratonovich type. If, on the other hand, the colored noise is smooth on the scale of particle relaxation, then the drift correction is the standard Stratonovich correction. If the noise is rough on this scale, then there is no drift correction. Strong (i.e., pathwise) techniques are used for the proof of the convergence theorems.

Item Type:Article
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Additional Information:© 2005 Society for Industrial and Applied Mathematics. Received by the editors June 24, 2004; accepted for publication (in revised form) November 23, 2004; published electronically June 3, 2005. This work was supported by the Engineering and Physical Sciences Research Council. The authors are grateful to D. Cai, P. R. Kramer, and J. C. Mattingly for useful suggestions. They are particularly grateful to J. M. Sancho for useful suggestions and for providing them with [27, 28].
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
Subject Keywords:white noise limits, Ornstein–Uhlenbeck process, Kraichnan model, Wong–Zakai theorem
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Other Numbering System NameOther Numbering System ID
Andrew StuartJ63
Issue or Number:1
Classification Code:AMS subject classifications. 60H10, 60H15, 60H30, 60G15
Record Number:CaltechAUTHORS:20170612-130002475
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78109
Deposited By: Tony Diaz
Deposited On:12 Jun 2017 20:59
Last Modified:15 Nov 2021 17:36

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