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

TL;DR

This paper investigates the asymptotic behavior of fractional Brownian motion with multivariate time within large convex domains, revealing how boundary smoothness influences persistence probabilities as the domain expands.

Contribution

It establishes the log-asymptotics of persistence probabilities for H-FBM in expanding convex sets, extending results to certain non-self-similar Gaussian processes.

Findings

Persistence probability decays as $(H - d + o(1)) imes ext{log} T$

Boundary smoothness affects asymptotic behavior

Generalizations to non-self-similar Gaussian processes

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.