A fuzzy set-valued stochastic framework for birth-and-growth process. Statistical aspects

The paper considers a particular family of fuzzy monotone set–valued stochastic processes. In order to investigate suitable α–level sets of such processes, a set–valued stochastic framework is proposed for the well–posedness of birth–and–growth process. A birth–and–growth model is rigorously defined as a suitable combination, involving Minkowski sum and 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. A decomposition theorem is established to characterize nucleation and growth. As a consequence, different consistent set–valued estimators are studied for growth processes. Moreover, the nucleation process is studied via the hitting function, and a consistent estimator of the nucleation hitting function is derived.