Necessary and sufficient conditions for the finiteness of the second moment of the measure of level sets

Abstract

For a smooth vectorial stationary Gaussian random field, $X:\Omega \times \mathbb {R}^{d}\to \mathbb {R}^{d}$, we provided necessary conditions to have a finite second moment for the number of roots of $X(t)-u$. Then, under a more restrictive hypothesis, some sufficient conditions were also given. The results were obtained using a method of proof inspired the one obtained by D. Geman for stationary Gaussian processes. Afterward, the same method is applied to the number of critical points of a scalar random field and to the level set of a vectorial process, $X:\Omega \times \mathbb {R}^{D}\to \mathbb {R}^{d}$, with $D>d$.