Singular Perturbations in Noisy Dynamical Systems

Consider a deterministic dynamical system in a domain containing a stable equilibrium, e.g., a particle in a potential well. The particle, independent of initial conditions, eventually reaches the bottom of the well. If however, the particle is subjected to white noise, due, e.g., to collisions with a population of smaller, lighter particles comprising the medium through which the particle travels, a dramatic difference in the behavior of the Brownian particle occurs. The particle will exit the well. The natural questions then are: how long will it take for it to exit and from where on the boundary of the domain of attraction of the deterministic equilibrium (the rim of the well) will it exit. We compute the mean first passage time to the boundary and the mean probabilities of the exit positions. When the noise is small each quantity satisfies a singularly perturbed deterministic boundary value problem. We treat the problem by the method of matched asymptotic expansions (MAE) and generalizations thereof. MAE has been used successfully to solve problems in many applications. However, there exist problems for which MAE does not suffice. Among these are problems exhibiting boundary layer resonance, i.e., the problem of ”spurious solutions”, which led some to conclude that this was ”the failure of MAE”. We present a physical argument and four mathematical arguments to modify or augment MAE to make it successful. Finally, we discuss applications of the theory. Introduction. I am honored and humbled by the award of the John von Neumann Lecture Prize by SIAM. This award is particularly meaningful to me, as four of the previous awardees were my teachers and my inspirations, including Kurt Friedrichs, Peter Lax, Jurgen Moser, and most importantly, my advisor Joe Keller. To have my name linked to theirs is a great honor indeed. Since two of these luminaries chose the subject of asymptotic expansions and applications thereof, for award talks, I have chosen to follow in their footsteps and speak on the same topic. Asymptotics is the study of the local behavior of a function. The function may be known a-priori or we may only have hints as to what it is, e.g., that it satisfies a differential equation and associated boundary and/or initial conditions. In this case we employ a perturbation method to solve the problem. One such problem concerns the effect of a small perturbation, e.g., noise, on a deterministic dynamical system. If the perturbation leads to only a small effect on the system we refer to this as a regular perturbation, whose result generally will be little noted nor long remembered. However, it is possible for the perturbation to have a dramatic effect, which is of far greater interest and is referred to as a singular perturbation. This can occur, e.g., if the perturbation is random. The differential equation This paper corresponds to the von Neumann Lecture Prize awarded in 2017