Large Deviations for One Dimensional Diffusions with a Strong Drift

Abstract

We derive a large deviation principle which describes the behaviour of a diffusion process with additive noise under the influence of a strong drift. Our main result is a large deviation theorem for the distribution of the end-point of a one-dimensional diffusion with drift $\theta b$ where $b$ is a drift function and $\theta$ a real number, when $\theta$ converges to $\infty$. It transpires that the problem is governed by a rate function which consists of two parts: one contribution comes from the Freidlin-Wentzell theorem whereas a second term reflects the cost for a Brownian motion to stay near a equilibrium point of the drift over long periods of time.