Scaling limit for the diffusion exit problem
Almada Monter, Sergio Angel
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A stochastic differential equation with vanishing martingale term is studied. Specifically, given a domain D, the asymptotic scaling properties of both the exit time from the domain and the exit distribution are considered under the additional (non-standard) hypothesis that the initial condition also has a scaling limit. Methods from dynamical systems are applied to get more complete estimates than the ones obtained by the probabilistic large deviation theory. Two situations are completely analyzed. When there is a unique critical saddle point of the deterministic system (the system without random effects), and when the unperturbed system escapes the domain D in finite time. Applications to these results are in order. In particular, the study of 2-dimensional heteroclinic networks is closed with these results and shows the existence of possible asymmetries. Also, 1-dimensional diffusions conditioned to rare events are further studied using these results as building blocks. The approach tries to mimic the well known linear situation. The original equation is smoothly transformed into a very specific non-linear equation that is treated as a singular perturbation of the original equation. The transformation provides a classification to all 2-dimensional systems with initial conditions close to a saddle point of the flow generated by the drift vector field. The proof then proceeds by estimates that propagate the small noise nature of the system through the non-linearity. Some proofs are based on geometrical arguments and stochastic pathwise expansions in noise intensity series.