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The fractional Brownian motion of index $0 < H < 1$, H-FBM, withd-dimensional time is considered on an expanding set TG, where G is a boundedconvex domain that contains 0 at its boundary. The main result: if 0 is a pointof smoothness of the boundary, then the log-asymptotics of probability thatH-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 notself-similar Gaussian processes with stationary increments are also considered.

Factional Brownian motion with multivariate time in a large convex area: persistence exponents