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

Abstract

Let I be a compact d-dimensional manifold, let X:I→ℛ be a Gaussian process with regular paths and let F_I(u), u∈ℛ, be the probability distribution function of sup _t∈IX(t).

We prove that under certain regularity and nondegeneracy conditions, F_I 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→ℛ^d is a random field and x is a fixed point in ℛ^d. We also give proofs for this kind of formulae, which have their own interest beyond the present application.