Unilateral small deviations of processes related to the fractional Brownian motion

Let x(s), s[set membership, variant]Rd be a Gaussian self-similar random process of index H. We consider the problem of log-asymptotics for the probability pT that x(s), x(0)=0 does not exceed a fixed level in a star-shaped expanding domain T[dot operator][Delta] as T-->[infinity]. We solve the problem of the existence of the limit, [theta]:=lim(-logpT)/(logT)D, T-->[infinity], for the fractional Brownian sheet x(s), s[set membership, variant][0,T]2 when D=2, and we estimate [theta] for the integrated fractional Brownian motion when D=1.