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Integration in a dynamical stochastic geometric framework. (English) Zbl 1264.60010

Summary: Motivated by the well-posedness of birth-and-growth processes, a stochastic geometric differential equation and, hence, a stochastic geometric dynamical system are proposed. In fact, a birth-and-growth process can be rigorously modeled as a suitable combination, involving the Minkowski sum and the Aumann integral, of two very general set-valued processes representing nucleation and growth dynamics, respectively. The simplicity of the proposed geometric approach allows to avoid problems of boundary regularities arising from an analytical definition of the front growth. In this framework, growth is generally anisotropic and, according to a mesoscale point of view, non-local, i.e., at a fixed time instant, the growth is the same at each point of the space.

MSC:

60D05 Geometric probability and stochastic geometry
53C65 Integral geometry
60H05 Stochastic integrals
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