The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments

Let X = {X(t), t ϵ R^N} be a centered Gaussian random field with stationary increments and X(0) = 0. For any compact rectangle T ⊂ R^N and u ϵ R, denote by A_u = {t ϵ T :X(t) ≥ u} the excursion set. Under X(·) ∈ C²(R^N) and certain regularity conditions, the mean Euler characteristic of A_u, denoted by E{ø(A_u)}, is derived. By applying the Rice method, it is shown that, as u→∞, the excursion probability P{suptϵT X(t) ≥ u} can be approximated by E{ø(A_u)} such that the error is exponentially smaller than E{ø(A_u)}. This verifies the expected Euler characteristic heuristic for a large class of Gaussian random fields with stationary increments.