The Annals of Applied Probability

Malliavin calculus approach to long exit times from an unstable equilibrium

Yuri Bakhtin and Zsolt Pajor-Gyulai

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


For a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. Using Malliavin calculus tools, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits.

Article information

Ann. Appl. Probab., Volume 29, Number 2 (2019), 827-850.

Received: October 2017
Revised: February 2018
First available in Project Euclid: 24 January 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65]

Vanishing noise limit unstable equilibrium exit problem polynomial decay Malliavin calculus


Bakhtin, Yuri; Pajor-Gyulai, Zsolt. Malliavin calculus approach to long exit times from an unstable equilibrium. Ann. Appl. Probab. 29 (2019), no. 2, 827--850. doi:10.1214/18-AAP1387.

Export citation


  • [1] Almada Monter, S. A. and Bakhtin, Y. (2011). Normal forms approach to diffusion near hyperbolic equilibria. Nonlinearity 24 1883–1907.
  • [2] Bakhtin, Y. (2008). Exit asymptotics for small diffusion about an unstable equilibrium. Stochastic Process. Appl. 118 839–851.
  • [3] Bakhtin, Y. (2010). Small noise limit for diffusions near heteroclinic networks. Dyn. Syst. 25 413–431.
  • [4] Bakhtin, Y. (2011). Noisy heteroclinic networks. Probab. Theory Related Fields 150 1–42.
  • [5] Bass, R. F. (2011). Stochastic Processes. Cambridge Series in Statistical and Probabilistic Mathematics 33. Cambridge Univ. Press, Cambridge.
  • [6] Champagnat, N. and Villemonais, D. (2016). Exponential convergence to quasi-stationary distribution and $Q$-process. Probab. Theory Related Fields 164 243–283.
  • [7] Day, M. V. (1995). On the exit law from saddle points. Stochastic Process. Appl. 60 287–311.
  • [8] Eizenberg, A. (1984). The exit distributions for small random perturbations of dynamical systems with a repulsive type stationary point. Stochastics 12 251–275.
  • [9] Freidlin, M. I. and Wentzell, A. D. (2012). Random Perturbations of Dynamical Systems, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 260. Springer, Heidelberg. Translated from the 1979 Russian original by Joseph Szücs.
  • [10] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [11] Kifer, Y. (1981). The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point. Israel J. Math. 40 74–96.
  • [12] Mikami, T. (1995). Large deviations for the first exit time on small random perturbations of dynamical systems with a hyperbolic equilibrium point. Hokkaido Math. J. 24 491–525.
  • [13] Nourdin, I. and Viens, F. G. (2009). Density formula and concentration inequalities with Malliavin calculus. Electron. J. Probab. 14 2287–2309.
  • [14] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.