May 2021 Atypical exit events near a repelling equilibrium
Yuri Bakhtin, Hong-Bin Chen
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Ann. Probab. 49(3): 1257-1285 (May 2021). DOI: 10.1214/20-AOP1479


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.


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Yuri Bakhtin. Hong-Bin Chen. "Atypical exit events near a repelling equilibrium." Ann. Probab. 49 (3) 1257 - 1285, May 2021.


Received: 1 November 2019; Revised: 1 June 2020; Published: May 2021
First available in Project Euclid: 7 April 2021

Digital Object Identifier: 10.1214/20-AOP1479

Primary: 60H07 , 60H10 , 60J60

Keywords: equidistribution , Exit problem , Malliavin calculus , polynomial decay , unstable equilibrium , Vanishing noise limit

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 3 • May 2021
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