Enea G. Bongiorno

Statistical aspects of fuzzy monotone set-valued stochastic processes. Application to birth-and-growth processes

The paper considers a particular family of fuzzy monotone set-valued stochastic processes. The proposed setting allows us to investigate suitable @a-level sets of such processes, modeling birth-and-growth processes. A decomposition theorem is established to characterize the nucleation and the growth. As a consequence, different consistent set-valued estimators are studied for growth process. Moreover, the nucleation process is studied via the hitting function, and a consistent estimator of the nucleation hitting function is derived.

INTEGRATION IN A DYNAMICAL STOCHASTIC GEOMETRIC FRAMEWORK

We propose a set–valued framework for the well–posedness of birth– and–growth process. Our 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 used geometrical approach leads us to avoid problems arising by 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, it is not local, i.e. for a fixed time instant, growth is the same at each space point.

A decomposition theorem for fuzzy set-valued random variables

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for which the Hukuhara difference

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