Abstract
We apply previous results on the pathwise exponential loss of memory of the initial condition for stochastic differential equations with small diffusion to the problem of the asymptotic distribution of the first exit times from an attracted domain. We show under general hypotheses that the suitably rescaled exit time converges in the zero-noise limit to an exponential random variable. Then we extend the results to an infinite-dimensional case obtained by adding a small random perturbation to a nonlinear heat equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. D. Ventzel and M. I. Freidlin, On small random perturbation of dynamical systems,Usp. Math. Nauk 25:3 (1970) [English transl.,Russ. Math. Surv. 25:1 (1970)].
A. D. Ventzel and M. I. Freidlin, Some problems concerning stability under small random perturbations,Theory Prob. Appl. 17(2):269 (1972).
A. D. Ventzel and M. I. Freidlin, Random perturbations of dynamical systems (Springer-Verlag, 1984).
M. Cassandro, A. Galves, E. Olivieri, and M. E. Vares, Metastable behaviour of stochastic dynamics: A pathwise approach,J. Stat. Phys. 35(5/6) (1984).
M. V. Day, On the exponential exit law in the small parameter exit problem,Stochastics 8:297 (1983).
A. Galves, E. Olivieri, and M. E. Vares, Metastability for a dynamical system subject to a small random perturbation,Ann. Prob. (1987).
F. Martinelli and E. Scoppola, Small random perturbation of dynamical systems: Exponential loss of memory of the initial conditions,Commun. Math. Phys. 120:25–69 (1988).
B. Faris and G. Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise,J. Phys. A: Math. Gen. 15:3025 (1982).
M. Cassandro, E. Olivieri, and P. Picco, Small random perturbations of infinite dimensional dynamical systems and nucleation,Ann. Inst. Henri Poincaré 44(4):343 (1986).
I. Guikhman and A. Skorokhod,Introduction à la théorie des processus aléatoires (MIR, 1980).
N. Chafee and E. Infante, A bifurcation problem for a nonlinear parabolic equation,Applicable Analysis 4:17 (1974).
M. I. Freidlin, private communication.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Martinelli, F., Olivieri, E. & Scoppola, E. Small random perturbations of finite- and infinite-dimensional dynamical systems: Unpredictability of exit times. J Stat Phys 55, 477–504 (1989). https://doi.org/10.1007/BF01041595
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01041595