## Advances in Applied Probability

- Adv. in Appl. Probab.
- Volume 48, Number 3 (2016), 712-725.

### On the capacity functional of excursion sets of Gaussian random fields on ℝ^{2}

#### Abstract

When a random field (*X*_{t},*t*∈ℝ^{2}) 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.

#### Article information

**Source**

Adv. in Appl. Probab., Volume 48, Number 3 (2016), 712-725.

**Dates**

First available in Project Euclid: 19 September 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1474296311

**Mathematical Reviews number (MathSciNet)**

MR3568888

**Zentralblatt MATH identifier**

1351.60063

**Subjects**

Primary: 60G15: Gaussian processes

Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G60: Random fields

**Keywords**

Capacity functional crossings excursion set Gaussian field growing circle method Rice formula second moment measure sweeping line method stereology stochastic geometry

#### Citation

Kratz, Marie; Nagel, Werner. On the capacity functional of excursion sets of Gaussian random fields on ℝ 2. Adv. in Appl. Probab. 48 (2016), no. 3, 712--725. https://projecteuclid.org/euclid.aap/1474296311