The Problem of Small Unilateral Deviations: the Existence of Decay Exponents

Abstract

Let x(s), s in R^d be a Gaussian self-similar random process of index H. We consider the problem of log-asymptotics for the probability p(T) that x(s), x(0)=0 does not exceed a fixed level in a star-shaped expanding domain TxG as T>>1. We solve the problem of the existence of the limit, theta:=lim (-log p(T))/(log T)^D, T>>1, for the fractional Brownian sheet x(s)on [0,T]^2 then D=2 and we estimate the theta for the integrated fractional Brownian motion then D=1.