Excursion probability of certain non-centered smooth Gaussian random fields

Let X={X(t),t∈T}X={X(t),t∈T} be a non-centered, unit-variance, smooth Gaussian random field indexed on some parameter space TT, and let Au(X,T)={t∈T:X(t)≥u}Au(X,T)={t∈T:X(t)≥u} be the excursion set. It is shown that, as u→∞u→∞, the excursion probability P{supt∈TX(t)≥u}P{supt∈TX(t)≥u} can be approximated by the expected Euler characteristic of Au(X,T)Au(X,T), denoted by E{χ(Au(X,T))}E{χ(Au(X,T))}, such that the error is super-exponentially small. The explicit formulae for E{χ(Au(X,T))}E{χ(Au(X,T))} are also derived for two cases: (i) TT is a rectangle and X−EXX−EX is stationary; (ii) TT is an NN-dimensional sphere and X−EXX−EX is isotropic.