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

  • Marie Kratz (a1) and Werner Nagel (a2)

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.

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* Postal address: ESSEC Business School, Avenue Bernard Hirsch BP 50105, Cergy-Pontoise 95021 cedex, France.
** Postal address: Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, D-07737 Jena, Germany. Email address: werner.nagel@uni-jena.de

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