Chord-distribution functions and Rice formulae. Application to random media.

Anne Estrade (1) , Ileana Iribarren (2) , Marie Kratz (3, 4, 5)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 Escuela de Matematicas [Caracas]
3 CES - Centre d'économie de la Sorbonne
4 ESSEC Business School
5 SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)

We consider an isotropic bi-phasic (pore and solid) medium, draw many lines through it, and see each line as a one-dimensional level-cut process with values 0 or 1 according to whether a regular stationary process $X$ is less or greater than a given level. The lengths of time intervals when $X$ is in a given phase are named chord functions. We are interested into obtaining information on chord functions in order to get most characterizations of the medium such as the volume fractions, the two-point correlation function or the specific surface area. Working with the Palm probability measure and using level crossings techniques, Rice methods in particular, allow to obtain the exact analytical formula of the chord-distribution function as well as the joint distribution function of two successive chords and the two-point correlation function. Also the ``Independent Interval Assumption'' is discussed.

Mots clés: Rice formula; chord; (up/down) crossing; Gaussian process; level-cut process; Palm probability; porous medium

Domaines: Probabilités [math.PR]

Anne Estrade, Ileana Iribarren, Marie Kratz. Chord-distribution functions and Rice formulae. Application to random media.. Extremes, 2012, 15, pp.333-352. ⟨hal-00161806⟩