The Exit Problem for Randomly Perturbed Dynamical Systems

Abstract

The cumulative effect on dynamical systems, of even very small random perturbations, may be considerable after sufficiently long times. For example, even if the corresponding deterministic system has an asymptotically stable equilibrium point, random effects can cause the trajectories of the system to leave any bounded domain with probability one. In this paper we consider the effect of small random perturbations of the type referred to as Gaussian white noise, on a (deterministic) dynamical system $\dot x = b(x)$. The vector $x(t)$ then becomes a stochastic process $x_\varepsilon (t)$ which satisfies the stochastic differential equation $dx_\varepsilon = b(x_\varepsilon )dt + \varepsilon \sigma (x_\varepsilon )dw$. Here $w(t)$ is the n dimensional Wiener process (Brownian motion), $b(x)$ is a vector field, $\sigma (x)$ is the diffusion matrix and $\varepsilon \ne 0$ is a small real parameter. We give the first complete formal solution of the following problem originally posed by Kolmogorov: Find the asymptotic expansion in $\varepsilon $ of (i) the probability distribution of the points on the boundary of a domain, where trajectories of the perturbed system first exit, and (ii) of the expected exit times. Our method is to relate the solutions of the above problems to the solutions of various singularly perturbed elliptic boundary value problems with turning points, whose solutions are then constructed asymptotically.