On the distribution of the maximum of a gaussian field with d parameters
Résumé
Let I be a compact d-dimensional manifold, let X:I\\to R be a Gaussian process with regular paths and let F_I(u), u\\in R, be the probability distribution function of sup_{t\\in I}X(t). We prove that under certain regularity and nondegeneracy conditions, F_I is a C^1-function and satisfies a certain implicit equation that permits to give bounds for its values and to compute its asymptotic behavior as u\\to +\\infty. 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\\to 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.