The first exit time of a Brownian motion from an unbounded convex domain

Abstract

Consider the first exit time $\tau_D$ of a $(d+1)$-dimensional Brownian motion from an unbounded open domain $D=\set{ (x, y) \in \R^{d+1} \dvtx y > f(x), x \in \R^d }$ starting at\vspace{0.5pt} $(x_0, f(x_0)+1)\in \R^{d+1}$ for some $x_0 \in \R^d$, where the function $f(x)$ on $\R^d$ is convex and $f(x) \to \infty$ as the Euclidean norm $|x| \to \infty$. Very general estimates for the asymptotics of $\log \pr{\tau_D >t}$ are given by using Gaussian techniques. In particular, for $f(x)=\exp\{ |x|^p\}$, $p >0$, \[ \lim_{t\to\infty} t^{-1} (\log t)^{2/p} \log \pr{\tau_D>t}=-j_\nu^2/2, \] where $\nu=(d-2)/2$ and $j_\nu$ is the smallest positive zero of the Bessel function $J_{\nu}$.