  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.

Probability;Other Condensed Matter