The First Exit Time of a Brownian Motion from the Minimum and Maximum Parabolic Domains

Consider a Brownian motion starting at an interior point of the minimum or maximum parabolic domains, namely, D(min) = {(x, y(1), y(2)) : ||x|| < min{(y(1) + 1)(1/p1), (y(2) + 1)(1/p2)}} and D(max) = {(x, y(1), y(2)) : ||x|| < max{(y(1) + 1)(1/p1), (y(2) + 1)(1/p2)}} in R(d+2), d >= 1, respectively, where ||.|| is the Euclidean norm in R(d), y(1), y(2) >= -1, and p(1), p(2) > 1. Let iota(Dmin) and iota(Dmax) denote the first times the Brownian motion exits from D(min) and D(max). Estimates with exact constants for the asymptotics of log P(iota(Dmin) > t) and log P(iota(Dmax) > t) are given as t -> infinity, depending on the relationship between p(1) and p(2), respectively. The proof methods are based on Gordon's inequality and early works of Li, Lifshits, and Shi in the single general parabolic domain case.