On the generalized geometric densities of random closed sets. An application to growth processes

In recent literature the authors have introduced a Delta formalism, á la Dirac, for the description of random closed sets of lower dimension with respect to the environment space d. Mean densities can be introduced for expected measures associated with such sets, with respect to the usual Lebesgue measure. In this paper we offer a review of the main results; in particular approximating sequences for the quoted mean densities are provided, that are of interest in the concrete estimation of mean densities of fibre processes, surface processes, etc. For time dependent random closed sets, as the ones describing the evolution of birth-and-growth processes (of interest for many models in material science and in biomedicine), the Delta formalism provides a natural framework for deriving evolution equations for mean densities at any (integer) Hausdorff dimension, in terms of the relevant kinetic parameters. In this context connections with the concepts of hazard functions, and spherical contact functions are presented.