On the capacity functional of excursion sets of Gaussian random fields on $\R^2$

TL;DR

This paper investigates the geometric properties of excursion sets of Gaussian random fields in two dimensions, extending one-dimensional results using stochastic geometry and Rice methods to analyze capacity functionals and boundary measures.

Contribution

It introduces new analysis of excursion set functionals for 2D Gaussian fields, expanding prior 1D results with advanced stochastic geometry techniques.

Findings

Extended capacity functional results to 2D Gaussian fields

Derived boundary length second moment measures

Applied Rice methods and stochastic geometry tools

Abstract

When a random field $(X_t, \ t\in {\mathbb R}^2)$ is thresholded on a given level $u$, the excursion set is given by its indicator $~1_{[u, \infty)}(X_t)$. 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.