Hitting probabilities of a class of Gaussian random fields

Let X = {X(t), t is an element of R-N} be an (N, d) -Gaussian random field whose components are independent copies of a centered Gaussian random field X-0. Under the assumption that the canonical metric root E(X-0(t) - X-0(s))(2) is commensurate with gamma(Sigma(N)(j=1) vertical bar t(j) - s(j)vertical bar(Hj)), where s = (s(1),..., s(N)), t = (t(1),..., t(N)) is an element of R-N, H-j is an element of (0, 1), j = 1, 2,..., N and gamma(r) is a non negative function with some mild conditions, upper and lower bounds on the hitting probabilities of X are obtained. To illustrate our results, several examples of Gaussian random fields are given. (C) 2016 Elsevier B.V. All rights reserved.