A Stochastic Geometric Framework for Dynamical Birth-and-Growth Processes: Related Statistical Analysis

A birth-and-growth model is rigorously defined as a suitable combination, involving the Minkowski sum and the Aumann integral, of two very general set-valued processes representing nucleation and growth respectively. The simplicity of the proposed geometrical approach let us avoid problems arising from an analytical definition of the front growth such as boundary regularities. In this framework, growth is generally anisotropic and, according to a mesoscale point of view, is not local, i.e. for a fixed time instant, growth is the same at each point space. The proposed setting allows us to investigate nucleation and growth processes also from a statistical point of view. Different consistent set-valued estimators for growth processes and for the nucleation hitting function are derived.