A central limit theorem for the Euler characteristic of a Gaussian excursion set

We study the Euler characteristic of an excursion set of a stationary isotropic Gaussian random field $X:\Omega\times\mathbb{R}^d\to\mathbb{R}$. Let us fix a level $u\in \R$ and let us consider the excursion set above $u$, $A(T,u)=\{t\in T:\,X(t)\ge u\}$ where $T$ is a bounded cube $\subset \R^d$. The aim of this paper is to establish a central limit theorem for the Euler characteristic of $A(T,u)$ as $T$ grows to $\R^d$, as conjectured by R. Adler more than ten years ago. The required assumption on $X$ is $C^3$ regularity of the trajectories, non degeneracy of the Gaussian vector $X(t)$ and derivatives at any fixed point $t\in \R^d$ as well as integrability on $\R^d$ of the covariance function and its derivatives. The fact that $X$ is $C^3$ is stronger than Geman's assumption traditionally used in dimension one. Nevertheless, our result extends what is known in dimension one to higher dimension. In that case, the Euler characteristic of $A(T,u)$ equals the number of up-crossings of $X$ at level $u$, plus eventually one if $X$ is above $u$ at the left bound of the interval $T$.