On the distribution of the maximum of a Gaussian field with d parameters

Let I be a compact d-dimensional manifold, let X:I.. be a Gaussian process with regular paths and let FI(u), u.., be the probability distribution function of sup.t.IX(t). We prove that under certain regularity and nondegeneracy conditions, FI is a C1-function and satisfies a certain implicit equation that permits to give bounds for its values and to compute its asymptotic behavior as u.+.. This is a partial extension of previous results by the authors in the case d=1. Our methods use strongly the so-called Rice formulae for the moments of the number of roots of an equation of the form Z(t)=x, where Z:I..d is a random field and x is a fixed point in .d. We also give proofs for this kind of formulae, which have their own interest beyond the present application.