Double extreme on joint sets for Gaussian random fields

For a centered Gaussian random field X={X(t),t∈RN}, let T1 and T2 be two compact sets in RN such that I=T1∩T2≠0̸ and denote by χ(Au(I)) the Euler characteristic of the excursion set Au(I)={t∈I:X(t)≥u}. We show that under certain smoothness and regularity conditions, as u→∞, the joint excursion probability P{supt∈T1X(t)≥u,sups∈T2X(s)≥u} can be approximated by the expected Euler characteristic E{χ(Au(I))} such that the error is super-exponentially small.