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.
Keywords: random closed set, stochastic geometry, birth-and-growth process, set-valued process, Aumann integral, Minkowski sum
@article{PS_2011__15__402_0,
author = {Aletti, Giacomo and Bongiorno, Enea G. and Capasso, Vincenzo},
title = {Integration in a dynamical stochastic geometric framework},
journal = {ESAIM: Probability and Statistics},
pages = {402--416},
year = {2011},
publisher = {EDP-Sciences},
volume = {15},
doi = {10.1051/ps/2010009},
mrnumber = {2870523},
zbl = {1264.60010},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/ps/2010009/}
}
TY - JOUR AU - Aletti, Giacomo AU - Bongiorno, Enea G. AU - Capasso, Vincenzo TI - Integration in a dynamical stochastic geometric framework JO - ESAIM: Probability and Statistics PY - 2011 SP - 402 EP - 416 VL - 15 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/ps/2010009/ DO - 10.1051/ps/2010009 LA - en ID - PS_2011__15__402_0 ER -
%0 Journal Article %A Aletti, Giacomo %A Bongiorno, Enea G. %A Capasso, Vincenzo %T Integration in a dynamical stochastic geometric framework %J ESAIM: Probability and Statistics %D 2011 %P 402-416 %V 15 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/ps/2010009/ %R 10.1051/ps/2010009 %G en %F PS_2011__15__402_0
Aletti, Giacomo; Bongiorno, Enea G.; Capasso, Vincenzo. Integration in a dynamical stochastic geometric framework. ESAIM: Probability and Statistics, Tome 15 (2011), pp. 402-416. doi: 10.1051/ps/2010009
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