Unilateral small deviations of processes related to the fractional Brownian motion

Let x ( s ) , s ∈ 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 T ⋅ Δ as T → ∞ . We solve the problem of the existence of the limit, θ ≔ lim ( − log p T ) / ( log T ) D , T → ∞ , for the fractional Brownian sheet x ( s ) , s ∈ [ 0 , T ] 2 when D = 2 , and we estimate θ for the integrated fractional Brownian motion when D = 1 .