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 ≠ 0{combining long solidus overlay} and denote by χ (_{A u} (I) ) the Euler characteristic of the excursion set _{A u} (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.