The First Time of a Brownian Motion from an Unbounded Convex Domain

Abstract

Consider the first exit time, ˝D of a (d + 1)-dimensional Brownian motion from an unbounded open domain D =

(x; y) 2 R d+1 : y > f(x); x 2 R d

starting at (x0; f(x0) + 1) 2 R d+1 for some x0 2 R d , where the function f(x) on R d is convex and f(x) ! 1 as the Euclidean norm j x j ! 1 . Very general estimates for the asymptotics of log P (˝D > t) are given by using Gaussian techniques. In particular, for f(x) = exp fj x j p g , p > 0, lim t !1 t