Unilateral small deviations of processes related to the fractional Brownian motion

Abstract

Let $x\left(s\right)$, $s\in R^d$ be a Gaussian self-similar random process of index $H$. We consider the problem of log-asymptotics for the probability $p_T$ that $x\left(s\right)$, $x\left(0\right)=0$ does not exceed a fixed level in a star-shaped expanding domain $T\cdot \Delta$ as $T\to \infty$. We solve the problem of the existence of the limit, $\theta ≔lim(-log{p}_T)/{(logT)}^D$, $T\to \infty$, for the fractional Brownian sheet $x\left(s\right)$, $s\in {[0,T]}^2$ when $D=2$, and we estimate $\theta$ for the integrated fractional Brownian motion when $D=1$.