The persistence exponents of Gaussian random fields connected by the Lamperti transform

Abstract

The (fractional) Brownian sheet is a simplest example of a Gaussian random field X whose covariance is the tensor product of a finite number (d) of nonnegative correlation functions of self-similar Gaussian processes. Let Y be the homogeneous Gaussian field obtained by applying to X the Lamperti transform, which involves the exponential change of time and the amplitude normalization to have unit variance. Under some assumptions, we prove the existence of the persistence exponents for both fields, X and Y, and find the relation between them. The exponent for X is the limit of ln1/P{X(t)<1, t in [0,T]^d}/(lnT)^d, T>>1. In terms of Y it has the form lim ln1/P{Y(t)<0,t in TG}/T^d:=Q, T>>1, where G is a suitable d-simplex and T is a similarity coefficient; G can be selected in form [0,c]^d if d=2 . The exponent Q exists for any continuous Gaussian homogeneous field Y with non-negative covariance when G is bounded region with a regular boundary. The problem of the existence and connection of exponents for X and Y was raised by Li and Shao [Ann. Probab. 32:1, 2004] and originally concerned the Brownian sheet.