On the capacity functional of excursion sets of Gaussian random fields on ℝ2

Abstract

When a random field (X_t,t∈ℝ²) is thresholded on a given level u, the excursion set is given by its indicator 1_{[u, ∞)}(X_t). The purpose of this work is to study functionals (as established in stochastic geometry) of these random excursion sets as, e.g. the capacity functional as well as the second moment measure of the boundary length. It extends results obtained for the one-dimensional case to the two-dimensional case, with tools borrowed from crossings theory, in particular, Rice methods, and from integral and stochastic geometry.