Abstract
The paper considers a particular family of fuzzy monotone set-valued stochastic processes. The proposed setting allows us to investigate suitable α -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.
Highlights
Nucleation and growth processes arise in several natural and technological applications such as, for example, solidification and phase–transition of materials, semiconductor crystal growth, biomineralization, and DNA replication
For what concerns the dynamical point of view, a parametric birth–and–growth process A birth–and–growth process is a random closed set (RaCS) family given by Θt =
T ∈ R+, where ΘTt n (Xn) is the RaCS obtained as the evolution up to time t > Tn of the germ born at time Tn in location Xn, according to some growth model
Summary
Nucleation and growth processes arise in several natural and technological applications (cf. [5, 6] and the references therein) such as, for example, solidification and phase–transition of materials, semiconductor crystal growth, biomineralization, and DNA replication (cf., e.g., [17]). This paper is an attempt to offer an original approach based on a purely geometric stochastic point of view in order to avoid regularity assumptions describing birth– and–growth processes. In view of this approach, we introduce different set–valued parametric estimators of the rate of growth of the process. They arise naturally from a decomposition via Minkowski sum and they are consistent as the observation window expands to the whole space.
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