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.