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We study the asymptotic behavior of a diffusion process with small diffusion in a domain $D$. This process is reflected at $\partial D$ with respect to a co-normal direction pointing inside $D$. Our asymptotic result is used to study the long time behavior of the solution of the corresponding parabolic PDE with Neumann boundary condition.

Small random perturbations may have a dramatic impact on the long time evolution of dynamical systems, and large deviation theory is often the right theoretical framework to understand these effects. At the core of the theory lies the minimization of an action functional, which in many cases of interest has to be computed by numerical means. Here we review the theoretical and computational aspects behind these calculations, and propose an algorithm that simplifies the geometric minimum action method introduced in [M. Heymann and E. Vanden-Eijnden, CPAM Vol. LXI, 1052-1117 (2008)] to minimize the action in the space of arc-length parametrized curves. We then illustrate this algorithm's capabilities by applying it to various examples from material sciences, fluid dynamics, atmosphere/ocean sciences, and reaction kinetics. In terms of models, these examples involve stochastic (ordinary or partial) differential equations with multiplicative or degenerate noise, Markov jump processes, and systems with fast and slow degrees of freedom, which all violate detailed balance, so that simpler computational methods are not applicable.

For piecewise expanding one-dimensional maps without periodic turning points we prove that isolated eigenvalues of small (random) perturbations of these maps are close to isolated eigenvalues of the unperturbed system. (Here ``eigenvalue'' means eigenvalue of the corresponding Perron-Frobenius operator acting on the space of functions of bounded variation.) This result applies e.g. to the approximation of the system by a finite state Markov chain and generalizes Ulam's conjecture about the approximation of the SBR invariant measure of such a map. We provide several simple examples showing that for maps with periodic turning points and for general multidimensional smooth hyperbolic maps isolated eigenvalues are typically unstable under random perturbations. Our main tool in the 1D case is a special technique for ``interchanging'' the map and the perturbation, developed in our previous paper, combined with a compactness argument.

In this paper we describe the long time behavior of solutions to quasi-linear parabolic equations with a small parameter at the second order term and the long time behavior of corresponding diffusion processes.