On the absolute continuity of random nodal volumes

We study the absolute continuity with respect to the Lebesgue measure of the distribution of the nodal volume associated with a smooth, nondegenerate and stationary Gaussian field (f(x),x.Rd). Under mild conditions, we prove that in dimension d.3, the distribution of the nodal volume has an absolutely continuous component plus a possible singular part. This singular part is actually unavoidable bearing in mind that some Gaussian processes have a positive probability to keep a constant sign on some compact domain. Our strategy mainly consists in proving closed Kac.Rice type formulas allowing one to express the volume of the set {f=0} as integrals of explicit functionals of (f,.f,Hess(f)) and next to deduce that the random nodal volume belongs to the domain of a suitable Malliavin gradient. The celebrated Bouleau.Hirsch criterion then gives conditions ensuring the absolute continuity.