Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-29T10:03:43.186Z Has data issue: false hasContentIssue false
  • English
  • Français

Integration in a dynamical stochastic geometric framework

Published online by Cambridge University Press:  05 January 2012

Giacomo Aletti ,
Enea G. Bongiorno  and
Vincenzo Capasso
Giacomo Aletti
Affiliation:
Department of Mathematics, University of Milan, via Saldini 50, 10133 Milan Italy. giacomo.aletti@unimi.it; enea.bongiorno@unimi.it; vincenzo.capasso@unimi.it
Enea G. Bongiorno
Affiliation:
Department of Mathematics, University of Milan, via Saldini 50, 10133 Milan Italy. giacomo.aletti@unimi.it; enea.bongiorno@unimi.it; vincenzo.capasso@unimi.it
Vincenzo Capasso
Affiliation:
Department of Mathematics, University of Milan, via Saldini 50, 10133 Milan Italy. giacomo.aletti@unimi.it; enea.bongiorno@unimi.it; vincenzo.capasso@unimi.it
Get access

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.

Keywords

Random closed setStochastic geometryBirth-and-growth processSet-valued processAumann integralMinkowski sum
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011 , pp. 402 - 416
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

G. Aletti, E.G. Bongiorno and V. Capasso, Statistical aspects of fuzzy monotone set-valued stochastic processes. application to birth-and-growth processes. Fuzzy Set. Syst. 160 (2009) 3140–3151.
Aletti, G. and Saada, D., Survival analysis in Johnson-Mehl tessellation. Stat. Infer. Stoch. Process. 11 (2008) 55–76. CrossRef
Aquilano, D., Capasso, V., Micheletti, A., Patti, S., Pizzocchero, L. and Rubbo, M., A birth and growth model for kinetic-driven crystallization processes, part i: Modeling. Nonlinear Anal. Real World Appl. 10 (2009) 71–92. CrossRef
J. Aubin and H. Frankowska, Set-valued Analysis. Birkhäuser, Boston Inc. (1990).
Barles, G., Soner, H.M. and Souganidiss, P.E., Front propagation and phase field theory. SIAM J. Control Optim. 31 (1993) 439–469. CrossRef
M. Burger, Growth fronts of first-order Hamilton-Jacobi equations. SFB Report 02-8, University Linz, Linz, Austria (2002).
M. Burger, V. Capasso and A. Micheletti, An extension of the Kolmogorov-Avrami formula to inhomogeneous birth-and-growth processes, in Math Everywhere. G. Aletti et al. Eds., Springer, Berlin (2007) 63–76.
Burger, M., Capasso, V. and Pizzocchero, L., Mesoscale averaging of nucleation and growth models. Multiscale Model. Simul. 5 (2006) 564–592 (electronic). CrossRef
V. Capasso (Ed.) Mathematical Modelling for Polymer Processing. Polymerization, Crystallization, Manufacturing. Mathematics in Industry 2, Springer-Verlag, Berlin (2003).
V. Capasso, On the stochastic geometry of growth, in Morphogenesis and Pattern Formation in Biological Systems. T. Sekimura, et al. Eds., Springer, Tokyo (2003) 45–58.
V. Capasso and D. Bakstein, An Introduction to Continuous-Time Stochastic Processes. Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston Inc. (2005).
Capasso, V. and Villa, E., Survival functions and contact distribution functions for inhomogeneous, stochastic geometric marked point processes. Stoch. Anal. Appl. 23 (2005) 79–96. CrossRef
C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions. Lecture Notes Math. 580, Springer-Verlag, Berlin (1977).
S.N. Chiu, Johnson-Mehl tessellations: asymptotics and inferences, in Probability, finance and insurance. World Sci. Publ., River Edge, NJ (2004) 136–149.
Chiu, S.N., Molchanov, I.S. and Quine, M.P., Maximum likelihood estimation for germination-growth processes with application to neurotransmitters data. J. Stat. Comput. Simul. 73 (2003) 725–732. CrossRef
N. Cressie, Modeling growth with random sets. In Spatial Statistics and Imaging (Brunswick, ME, 1988). IMS Lecture Notes Monogr. Ser. 20, Inst. Math. Statist., Hayward, CA (1991) 31–45.
D.J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes. Probability and its Applications, I, 2nd edition, Springer-Verlag, New York (2003).
N. Dunford and J.T. Schwartz, Linear Operators. Part I. Wiley Classics Library, John Wiley & Sons Inc., New York (1988).
Erhardsson, T., Refined distributional approximations for the uncovered set in the Johnson-Mehl model. Stoch. Proc. Appl. 96 (2001) 243–259. CrossRef
Frost, H.J. and Thompson, C.V., The effect of nucleation conditions on the topology and geometry of two-dimensional grain structures. Acta Metallurgica 35 (1987) 529–540. CrossRef
E. Giné, M.G. Hahn and J. Zinn, Limit theorems for random sets: an application of probability in Banach space results. In Probability in Banach Spaces, IV (Oberwolfach, 1982). Lecture Notes Math. 990, Springer, Berlin (1983) 112–135.
Herrick, J., Jun, S., Bechhoefer, J. and Bensimon, A., Kinetic model of DNA replication in eukaryotic organisms. J. Mol. Biol. 320 (2002) 741–750. CrossRef
Hiai, F. and Umegaki, H., Integrals, conditional expectations, and martingales of multivalued functions. J. Multivariate Anal. 7 (1977) 149–182. CrossRef
Himmelberg, C.J., Measurable relations. Fund. Math. 87 (1975) 53–72.
S. Li, Y. Ogura and V. Kreinovich, Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables. Kluwer Academic Publishers Group, Dordrecht (2002).
G. Matheron, Random Sets and Integral Geometry, John Wiley & Sons, New York-London-Sydney (1975).
A. Micheletti, S. Patti and E. Villa, Crystal growth simulations: a new mathematical model based on the Minkowski sum of sets, in Industry Days 2003-2004 The MIRIAM Project 2, D. Aquilano et al. Eds., Esculapio, Bologna (2005) 130–140.
I.S. Molchanov, Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, Chichester (1997).
Molchanov, I.S. and Chiu, S.N., Smoothing techniques and estimation methods for nonstationary Boolean models with applications to coverage processes. Biometrika 87 (2000) 265–283. CrossRef
Møller, J., Random Johnson-Mehl tessellations. Adv. Appl. Prob. 24 (1992) 814–844. CrossRef
Møller, J., Generation of Johnson-Mehl crystals and comparative analysis of models for random nucleation. Adv. Appl. Prob. 27 (1995) 367–383. CrossRef
Møller, J. and Sørensen, M., Statistical analysis of a spatial birth-and-death process model with a view to modelling linear dune fields. Scand. J. Stat. 21 (1994) 1–19.
Rådström, H., An embedding theorem for spaces of convex sets. Proc. Am. Math. Soc. 3 (1952) 165–169. CrossRef
J. Serra, Image Analysis and Mathematical Morphology. Academic Press Inc., London (1984).
Shoumei, L. and Aihong, R., Representation theorems, set-valued and fuzzy set-valued Ito integral. Fuzzy Set. Syst. 158 (2007) 949–962.
D. Stoyan, W.S. Kendall and J. Mecke, Stochastic Geometry and its Applications. 2nd edition, John Wiley & Sons Ltd., Chichester (1995).