Let \(b_\gamma(t)\), \(b_\gamma(0)= 0\), be a fractional Brownian motion, i.e., a Gaussian process with the structure function \(E(|b_\gamma(t)- b_\gamma(s)|)^2=|t- s|^\gamma\), \(0<\gamma<2\). The author establishes the following results:
1. The maximum \(M_t\) of a fractional Brownian motion on \([0,t]\) obeys the asymptotics \(\ln P\{M_t< 1\}= D\ln t^{-1}(1+ O((\ln t)^{-1/2}))\), \(t\to\infty\), where \(D= 1-\gamma/2\).
2. If \(G\) is a bounded convex region on \(\mathbb{R}^d\) which contains a vicinity of the point \(t= 0\), and \(M(G)= \sup\{b_\gamma(t), t\in G\}\), then \(\ln P\{M(TG)< 1\}= -d\ln T(1+ O(\ln T)^{-1/2})\), \(T\to\infty\), where \(TG\) is a region similar to \(G\) and has similarity coefficient \(T\).