The Persistence Exponents of Gaussian Random Fields Connected by the Lamperti Transform

Abstract

The (fractional) Brownian sheet is the 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. We consider the homogeneous Gaussian field $Y$ obtained by applying to X the exponential change of time (more precisely, the Lamperti transform). 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 any random function $Z$ is $\psi (T)$ if lim $\ln P(Z({\mathbf{t}}) \le ,{\mathbf{t}} \in TG)/\psi (T) = - 1$, $T > > {1}$, where $G$ is a d-dimensional region containing 0, and $T$ is a similarity coefficient. The considered problem was raised by Li and Shao [Ann. Probab. 32:1, 2004] and originally concerned the Brownian sheet.