ATYPICAL EXIT EVENTS NEAR A REPELLING EQUILIBRIUM

Yuri Bakhtin, Hong Bin Chen

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Pages (from-to)1257-1285
Number of pages29
JournalAnnals of Probability
Volume49
Issue number3
DOIs
StatePublished - May 2021

Keywords

  • equidistribution
  • exit problem
  • Malliavin calculus
  • polynomial decay
  • unstable equilibrium
  • Vanishing noise limit

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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