Doob's Conditioned Diffusions and their Lifetimes

Abstract

We study the lifetime of a conditioned diffusion (or $h$-path) on a bounded $C^\infty$ domain $G$ in $\mathbb{R}^d$. Making use of results of Donsker and Varadhan, we show that the tail of the distribution of the lifetime decays exponentially; in fact, the decay constant is the same as that for the exponential decay of the tail of the distribution of the first time the unconditioned diffusion exits $G$. In the case of Brownian motion and bounded domains (not necessarily $C^\infty$) we describe some sufficient conditions to ensure the previously described asymptotic results hold here too.