  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 minimisers 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 only depends on the mean and coincides with the
$\Gamma$-limit of the 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.
