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Consider the rst exit time, D of a (d + 1)-dimensional Brownian motion from an unbounded open domain D = (x;y)2Rd+1 : y > f(x);x2Rd starting at (x0;f(x0) + 1)2 Rd+1 for some x0 2 Rd, where the function f(x) on Rd is convex and f(x)! 1 as the Euclidean norm jxj ! 1. Very general estimates for the asymptotics of logP ( D > t) are given by using Gaussian techniques. In particular, for f(x) = expfjxjpg, p > 0,

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