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Gaussian Approximations for Transition Paths in Brownian Dynamics

Lu, Yulong and Stuart, Andrew and Weber, Hendrik (2017) Gaussian Approximations for Transition Paths in Brownian Dynamics. SIAM Journal on Mathematical Analysis, 49 (4). pp. 3005-3047. ISSN 0036-1410. doi:10.1137/16M1071845.

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This paper is concerned with transition paths within the framework of the overdamped Langevin dynamics model of chemical reactions. We aim to give an efficient description of typical transition paths in the small temperature regime. We adopt a variational point of view and seek the best Gaussian approximation, with respect to Kullback--Leibler divergence, of the non-Gaussian distribution of the diffusion process. We interpret the mean of this Gaussian approximation as the “most likely path,” and the covariance operator as a means to capture the typical fluctuations around this most likely path. We give an explicit expression for the Kullback--Leibler divergence in terms of the mean and the covariance operator for a natural class of Gaussian approximations and show the existence of minimizers for the variational problem. Then the low temperature limit is studied via Γ-convergence of the associated variational problem. The limiting functional consists of two parts: The first part depends only on the mean and coincides with the Γ-limit of the rescaled Freidlin--Wentzell rate functional. The second part depends on both the mean and the covariance operator and is minimized if the dynamics are given by a time-inhomogenous Ornstein--Uhlenbeck process found by linearization of the Langevin dynamics around the Freidlin--Wentzell minimizer.

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Additional Information:© 2017 Society for Industrial and Applied Mathematics. Submitted: 22 April 2016; Accepted: 03 March 2017; Published online: 10 August 2017. The work of the first author was supported by the EPSRC as part of the MASDOC DTC at the University of Warwick, through grant EP/HO23364/1. The work of the second author was supported by DARPA, the EPSRC, and the ONR. The work of the third author was supported by the EPSRC and by the Royal Society. The authors are grateful to Frank Pinski for helpful discussions and insights.
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)EP/HO23364/1
Defense Advanced Research Projects Agency (DARPA)UNSPECIFIED
Office of Naval Research (ONR)UNSPECIFIED
Subject Keywords:transition path, Kullback--Leibler approximation, Onsager--Machlup functional, large deviations, gamma-convergence
Issue or Number:4
Classification Code:AMS Subject Headings: 28C20, 60G15, 60F10
Record Number:CaltechAUTHORS:20170921-105427126
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:81678
Deposited By: Tony Diaz
Deposited On:21 Sep 2017 18:04
Last Modified:15 Nov 2021 19:45

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