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Tails of exit times from unstable equilibria on the line

Part of: Markov processes Stochastic analysis Random dynamical systems

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: bakhtin@cims.nyu.edu
*Postal address: Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, NY, 10012, USA. Email: bakhtin@cims.nyu.edu
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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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References

Almada, S. andBakhtin, Y. (2011). Normal forms approach to diffusion near hyperbolic equilibria. Nonlinearity 24, 1883–1907.Google Scholar
Bakhtin, Y. (2010). Small noise limit for diffusions near heteroclinic networks. Dynam. Syst. 25, 413–431.10.1080/14689367.2010.482520CrossRefGoogle Scholar
Bakhtin, Y. (2011). Noisy heteroclinic networks. Prob. Theory Relat. Fields 150, 1–42.CrossRefGoogle Scholar
Bakhtin, Y. andPajor-Gyulai, Z. (2019). Malliavin calculus approach to long exit times from an unstable equilibrium. Ann. Appl. Prob. 29, 827–850.10.1214/18-AAP1387CrossRefGoogle Scholar
Bass, R. F. (2011). Stochastic Processes (Camb. Ser. Statist. Probab. Math. 33). Cambridge University Press.Google Scholar
Champagnat, N. andVillemonais, D. (2016). Exponential convergence to quasi-stationary distribution and Q-process. Prob. Theory Relat. Fields 164, 243–283.10.1007/s00440-014-0611-7CrossRefGoogle Scholar
Eizenberg, A. (1984). The exit distributions for small random perturbations of dynamical systems with a repulsive type stationary point. Stochastics 12, 251–275.CrossRefGoogle Scholar
Karatzas, I. andShreve, S. (1991). Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics. Springer, New York.Google Scholar