Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields

Abstract

Let X = {X(t),t∈ R N } be a Gaussian random field with values in R d defined by X(t )= (X1(t),...,Xd(t)), where X1,...,Xd are independent copies of a centered Gaussian random field X0. Under certain general conditions on X0, we study the hitting probabilities of X and determine the Hausdorff dimension of the inverse image X �1 (F ), where F ⊆ R d is a nonrandom Borel set. The class of Gaussian random fields that satisfy our conditions includes not only fractional Brownian motion and the Brownian sheet, but also such anisotropic fields as fractional Brownian sheets, solutions to stochastic heat equation driven by space-time white noise and the operator-scaling Gaussian random fields with stationary increments constructed �