Atypical exit events near a repelling equilibrium

TL;DR

This paper analyzes the asymptotic behavior of exit probabilities and distributions near a repelling equilibrium under small noise, revealing polynomial decay and detailed structure of atypical exit events.

Contribution

It provides a novel polynomial asymptotic analysis of exit probabilities and characterizes the limiting distributions for atypical exits near a repelling equilibrium.

Findings

Exit probability decays polynomially with noise strength.

Limiting distributions for atypical exits are absolutely continuous with respect to volume measures.

Contrasts with large deviation theory where limits are point masses.

Abstract

We consider exit problems for small white noise perturbations of a dynamical system generated by a vector field, and a domain containing a critical point with all positive eigenvalues of linearization. We prove that, in the vanishing noise limit, the probability of exit through a generic set on the boundary is asymptotically polynomial in the noise strength, with exponent depending on the mutual position of the set and the flag of the invariant manifolds associated with the top eigenvalues. Furthermore, we compute the limiting exit distributions conditioned on atypical exit events of polynomially small probability and show that the limits are Radon--Nikodym equivalent to volume measures on certain manifolds that we construct. This situation is in sharp contrast with the large deviation picture where the limiting conditional distributions are point masses.