Integration in a dynamical stochastic geometric framework

Department of Mathematics, University of Milan, via Saldini 50, 10133 Milan Italy. giacomo.aletti@unimi.it; enea.bongiorno@unimi.it; vincenzo.capasso@unimi.it

Received: 14 July 2009
Revised: 26 November 2009
Revised: 25 February 2010

Abstract

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, is non local, i.e. at a fixed time instant, growth is the same at each point of the space.