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

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}$}$,denoteby$A_u={t\in T:X(t)\geq u}$theexcursionset\mathrm{.}Under$X(\cdot)\in C^2({$\mathbb{R}$}^N)$ and certain regularity conditions, the mean Euler characteristic of $A_u$, denoted by $\mathbb{E}\{\phi (A_u)\}$, is derived. By applying the Rice method, it is shown that, as $u\to \mathrm{\infty }$, the excursion probability $\mathbb{P}\{{\sup }_{t\in T}X(t)\ge u\}$ can be approximated by $\mathbb{E}\{\phi (A_u)\}$ such that the error is exponentially smaller than $\mathbb{E}\{\phi (A_u)\}$. This verifies the expected Euler characteristic heuristic for a large class of Gaussian random fields with stationary increments.