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Tails of exit times from unstable equilibria on the line

Published online by Cambridge University Press:  16 July 2020

Yuri Bakhtin*
New York University
Zsolt Pajor-Gyulai*
New York University
*Postal address: Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, NY, 10012, USA. Email:
*Postal address: Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, NY, 10012, USA. Email:


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.

Research Papers
© Applied Probability Trust 2020

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