Mathematical Approaches to the Problem of Noise-Induced Exit

Abstract

We provide an overview of some of the methods that have been used to study sub-large deviations phenomena in the small noise exit problem. Both formal asymptotic methods and rigorous probabilistic methods are considered and compared.

The exit from a metastable state: Concentration of the exit point distribution on the low energy saddle points, part 1

“…They are mainly three kinds of approaches to study where and how the law of X τ Ω concentrates on ∂Ω when h → 0. We refer to [5] for a comprehensive review of the literature. The first approach one is based on formal computations: the concentration of the law of X τ Ω on arg min ∂Ω f in the small temperature regime (h → 0) has been studied in [34] when ∂ n f > 0 on ∂Ω and in [37,42] when considering also the case when ∂ n f = 0 on ∂Ω.…”

“…Here and in the following, the subscript x indicates that the stochastic process starts from x ∈ R d : X 0 = x. In words, (5) means that if X 0 is distributed according to ν h , then for all t > 0, X t is still distributed according to ν h conditionally on X s ∈ Ω for all s ∈ (0, t). We have the following results from [28]:…”

On the behavior of diffusion processes with traps

“…The time to exit is of order exp(V k (x k )/ε). Even if the minimum is not achieved in a single point, there are cases when the exit from the domain is well understood (e.g., in the simplest example when v is sphericaly symmetric and D k is a ball around O k , also see [1] and references there). We'll simply assume that for each compact K ⊂ D k , the exit time (appropriately re-scaled) and the exit location have limiting distributions that do not depend on the starting point within K, as in the case of a single minimum for the quasi-potential and in the symmetric case mentioned above.…”

Exit Event from a Metastable State and Eyring-Kramers Law for the Overdamped Langevin Dynamics

“…Finally, let us mention [20,21,31,37,46,49,50] for a study of the asymptotic behaviour in the limit h → 0 of λ h and u h (see Proposition 2 below). The reader can also refer to [19] for a review of the different techniques used to study the asymptotic behaviour of X τΩ when h → 0 and to [2] for a review of the different techniques used to study the asymptotic behaviour of τ Ω when h → 0.…”