  The fractional Brownian motion of index $0 < H < 1$, H-FBM, with
d-dimensional time is considered on an expanding set TG, where G is a bounded
convex domain that contains 0 at its boundary. The main result: if 0 is a point
of smoothness of the boundary, then the log-asymptotics of probability that
H-FBM does not exceed a fixed positive level in TG is $(H - d + o(1)) \log T$,
$T\to\infty$. Some generalizations of this result to isotropic but not
self-similar Gaussian processes with stationary increments are also considered.
