The Mean Euler Characteristic and Excursion Probability of Gaussian Random Fields with Stationary Increments

Let $X = {X(t), t\in \R^{N}}$ be a centered Gaussian random field with stationary increments and let $T \subset \R^N$ be a compact rectangle. Under $X(\cdot) \in C^2(\R^N)$ and certain additional regularity conditions, the mean Euler characteristic of the excursion set $A_u = {t\in T: X(t)\geq u}$, denoted by $\E{\varphi(A_u)}$, is derived. By applying the Rice method, it is shown that, as $u \to \infty$, the excursion probability $\P{\sup_{t\in T} X(t) \geq u}$ can be approximated by $\E{\varphi(A_u)}$ such that the error is exponentially smaller than $\E{\varphi(A_u)}$. This verifies the expected Euler characteristic heuristic for a large class of Gaussian random fields with stationary increments.