Hitting probabilities of a class of Gaussian random fields

Let X={X(t),t∈RN}X={X(t),t∈RN} be an (N,d)(N,d)-Gaussian random field whose components are independent copies of a centered Gaussian random field X0X0. Under the assumption that the canonical metric E(X0(t)−X0(s))2 is commensurate with γ(∑j=1N∣tj−sj∣Hj), where s=(s1,…,sN),t=(t1,…,tN)∈RN,Hj∈(0,1),j=1,2,…,Ns=(s1,…,sN),t=(t1,…,tN)∈RN,Hj∈(0,1),j=1,2,…,N and γ(r)γ(r) is a non-negative function with some mild conditions, upper and lower bounds on the hitting probabilities of XX are obtained. To illustrate our results, several examples of Gaussian random fields are given.