Gaussian Approximations for Transition Paths in Brownian Dynamics

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 $\Gamma$-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 $\Gamma$-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.