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

Abstract: Let $X={X(t),t\in {\mathbb{R}}^N}$ be a centered Gaussian random field with stationary increments and $X(0)=0$. For any compact rectangle $T\subset {\mathbb{R}}^N$ and $u\in {\mathbb{R}}$, denote by $A_u={t\in T:X(t)\geq u}$ the excursion set. Under $X(\cdot)\in C^2({\mathbb{R}}^N)$ and certain regularity conditions, the mean Euler characteristic of $A_u$, denoted by ${\mathbb{E}}{\varphi(A_u)}$, is derived. By applying the Rice method, it is shown that, as $u\to\infty$, the excursion probability ${\mathbb{P}}{\sup_{t\in T}X(t)\geq u}$ can be approximated by ${\mathbb{E}}{\varphi(A_u)}$ such that the error is exponentially smaller than ${\mathbb{E}}{\varphi(A_u)}$. This verifies the expected Euler characteristic heuristic for a large class of Gaussian random fields with stationary increments.