The mean euler characteristic and excursion probability of gaussian random fields with stationary increments

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