Unilateral small deviations of processes related to the fractional Brownian motion

Let x(s)x(s), s∈Rds∈Rd be a Gaussian self-similar random process of index HH. We consider the problem of log-asymptotics for the probability pTpT that x(s)x(s), x(0)=0x(0)=0 does not exceed a fixed level in a star-shaped expanding domain T⋅ΔT⋅Δ as T→∞T→∞. We solve the problem of the existence of the limit, θ≔lim(−logpT)/(logT)Dθ≔lim(−logpT)/(logT)D, T→∞T→∞, for the fractional Brownian sheet x(s)x(s), s∈[0,T]2s∈[0,T]2 when D=2D=2, and we estimate θθ for the integrated fractional Brownian motion when D=1D=1.