Exit time asymptotics for small noise stochastic delay differential equations

TL;DR

This paper studies the exit times of small noise stochastic delay differential equations near stable equilibria or periodic orbits, providing asymptotic estimates and establishing a uniform sample path large deviation principle.

Contribution

It introduces a uniform sample path large deviation principle for SDDEs with history-dependent coefficients, extending large deviation theory to infinite-dimensional stochastic delay systems.

Findings

Asymptotic estimates for exit times as noise vanishes

Proof of a uniform sample path large deviation principle for SDDEs

Methodology applicable to broader classes of infinite-dimensional stochastic equations

Abstract

Dynamical system models with delayed dynamics and small noise arise in a variety of applications in science and engineering. In many applications, stable equilibrium or periodic behavior is critical to a well functioning system. Sufficient conditions for the stability of equilibrium points or periodic orbits of certain deterministic dynamical systems with delayed dynamics are known and it is of interest to understand the sample path behavior of such systems under the addition of small noise. We consider a small noise stochastic delay differential equation (SDDE) with coefficients that depend on the history of the process over a finite delay interval. We obtain asymptotic estimates, as the noise vanishes, on the time it takes a solution of the stochastic equation to exit a bounded domain that is attracted to a stable equilibrium point or periodic orbit of the corresponding deterministic equation. To obtain these asymptotics, we prove a sample path large deviation principle (LDP) for the SDDE that is uniform over initial conditions in bounded sets. The proof of the uniform sample path LDP uses a variational representation for exponential functionals of strong solutions of the SDDE. We anticipate that the overall approach may be useful in proving uniform sample path LDPs for a broad class of infinite-dimensional small noise stochastic equations.