Methods of geometric measure theory in stochastic geometry

All the results of the present thesis have been obtained facing problems related to the study of the so called birth-and-growth stochastic processes, relevant in several real applications, like crystallization processes, tumour growth, angiogenesis, etc. We have introduced a Delta formalism, à la Dirac-Schwartz, for the description of random measures associated with random closed sets in R^d of lower dimensions, such that the usual Dirac delta at a point follows as particular case, in order to provide a natural framework for deriving evolution equations for mean densities at integer Hausdorff dimensions in terms of the relevant kinetic parameters associated to a given birth-and-growth process. In this context connections with the concepts of hazard functions and spherical contact distribution functions, together with local Steiner formulas at first order have been studied and, under suitable general conditions on the resulting random growing set, we may write evolution equations of the mean volume density in terms of the growing rate and of the mean surface density. To this end we have introduced definitions of discrete, continuous and absolutely continuous random closed set, which extend the standard well known definitions for random variables. Further, since in many real applications such as fibre processes, n-facets of random tessellations several problems are related to the estimation of such mean densities, in order to face such problems in the general setting of spatially inhomogeneous processes, we have analyzed an approximation of mean densities for sufficiently regular random closed sets, such that some known results in literature follow as particular cases.