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Tails of exit times from unstable equilibria on the line

Part of: Markov processes Random dynamical systems Stochastic analysis

Published online by Cambridge University Press:  16 July 2020

Yuri Bakhtin  and
Zsolt Pajor-Gyulai
Yuri Bakhtin*
Affiliation:
New York University
Zsolt Pajor-Gyulai*
Affiliation:
New York University
*
*Postal address: Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, NY, 10012, USA. Email: [email protected]
*Postal address: Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, NY, 10012, USA. Email: [email protected]
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Abstract

For a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. We give a revealing elementary proof of a result proved earlier using heavy machinery from Malliavin calculus. In particular, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits. We also discuss our program on rare transitions in noisy heteroclinic networks.

Keywords

vanishing noise limitunstable equilibriumexit problempolynomial decay

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 60J60: Diffusion processes 37H10: Generation, random and stochastic difference and differential equations
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 2 , June 2020 , pp. 477 - 496
Copyright
© Applied Probability Trust 2020

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