Cristallization processes: ergodic properties and statistical inference

Abstract

We consider a birth and growth model with germs being born according to a Poisson point process whose intensity measure is invariant under translations in space. The germs can be born in unoccupied space and then start growing until they occupy the available space. In this general framework, the crystallization process can be characterized by a random field which, for any point in the state space, assigns the first time at which this point is reached by a crystal. Under general conditions on the growth speed and geometrical shape of free crystals, the random field has been proved to be mixing in the sense of ergodic theory and absolute regularity coefficients can be estimated. These mixing properties are applied here to study almost sure convergence and asymptotic normality of some additive functionals involved in intensity measure parameters estimation. References: 1) Yu. Davydov, A. Illig, Ergodic properties of geometrical crystallization processes C. R., Math., Acad. Sci. Paris 345, No. 10, 583-586 (2007) 2) Yu. Davydov, A. Illig, Ergodic properties of geometrical crystallization processes arXiv:math/0610966 3) Yu. Davydov, A. Illig, Ergodic properties of crystallization processes Journal of Mathematical Sciences, V. 163, 4, pp. 375-381 (2009)