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

Abstract

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.

10 pages