The tail of the maximum of smooth Gaussian fields on fractal sets

Abstract

We study the probability distribution of the maximum $M_S$ of a smooth stationary Gaussian field defined on a fractal subset $S$ of ${\mathbb{R}}^n$. Our main result is the equivalent of the asymptotic behavior of the tail of the distribution $\P (M_S>u)$ as $u\to +\infty .$ The basic tool is Rice formula for the moments of the number of local maxima of a random field.