# Integration in a dynamical stochastic geometric framework

Giacomo Aletti; Enea G. Bongiorno; Vincenzo Capasso

ESAIM: Probability and Statistics (2011)

- Volume: 15, page 402-416
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topAletti, Giacomo, Bongiorno, Enea G., and Capasso, Vincenzo. "Integration in a dynamical stochastic geometric framework." ESAIM: Probability and Statistics 15 (2011): 402-416. <http://eudml.org/doc/277142>.

@article{Aletti2011,

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.},

author = {Aletti, Giacomo, Bongiorno, Enea G., Capasso, Vincenzo},

journal = {ESAIM: Probability and Statistics},

keywords = {random closed set; stochastic geometry; birth-and-growth process; set-valued process; Aumann integral; Minkowski sum},

language = {eng},

pages = {402-416},

publisher = {EDP-Sciences},

title = {Integration in a dynamical stochastic geometric framework},

url = {http://eudml.org/doc/277142},

volume = {15},

year = {2011},

}

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

PB - EDP-Sciences

VL - 15

SP - 402

EP - 416

AB - 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.

LA - eng

KW - random closed set; stochastic geometry; birth-and-growth process; set-valued process; Aumann integral; Minkowski sum

UR - http://eudml.org/doc/277142

ER -

## References

top- [1] 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. Zbl1209.62213MR2567098
- [2] G. Aletti and D. Saada, Survival analysis in Johnson-Mehl tessellation. Stat. Infer. Stoch. Process.11 (2008) 55–76. Zbl1148.62080MR2357553
- [3] D. Aquilano, V. Capasso, A. Micheletti, S. Patti, L. Pizzocchero and M. Rubbo, A birth and growth model for kinetic-driven crystallization processes, part i: Modeling. Nonlinear Anal. Real World Appl.10 (2009) 71–92. Zbl1154.82320MR2451692
- [4] J. Aubin and H. Frankowska, Set-valued Analysis. Birkhäuser, Boston Inc. (1990). Zbl1168.49014MR1048347
- [5] G. Barles, H.M. Soner and P.E. Souganidiss, Front propagation and phase field theory. SIAM J. Control Optim.31 (1993) 439–469. Zbl0785.35049MR1205984
- [6] M. Burger, Growth fronts of first-order Hamilton-Jacobi equations. SFB Report 02-8, University Linz, Linz, Austria (2002).
- [7] 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. Zbl1203.60127MR2281425
- [8] M. Burger, V. Capasso and L. Pizzocchero, Mesoscale averaging of nucleation and growth models. Multiscale Model. Simul. 5 (2006) 564–592 (electronic). Zbl1118.49023MR2247763
- [9] V. Capasso (Ed.) Mathematical Modelling for Polymer Processing. Polymerization, Crystallization, Manufacturing. Mathematics in Industry 2, Springer-Verlag, Berlin (2003). Zbl1004.00015MR1964653
- [10] 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.
- [11] V. Capasso and D. Bakstein, An Introduction to Continuous-Time Stochastic Processes. Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston Inc. (2005). Zbl1078.60001MR2102925
- [12] V. Capasso and E. Villa, Survival functions and contact distribution functions for inhomogeneous, stochastic geometric marked point processes. Stoch. Anal. Appl.23 (2005) 79–96. Zbl1068.60016MR2123945
- [13] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions. Lecture Notes Math. 580, Springer-Verlag, Berlin (1977). Zbl0346.46038MR467310
- [14] S.N. Chiu, Johnson-Mehl tessellations: asymptotics and inferences, in Probability, finance and insurance. World Sci. Publ., River Edge, NJ (2004) 136–149. MR2189203
- [15] S.N. Chiu, I.S. Molchanov and M.P. Quine, Maximum likelihood estimation for germination-growth processes with application to neurotransmitters data. J. Stat. Comput. Simul.73 (2003) 725–732. Zbl1042.62016MR2009433
- [16] 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. Zbl0769.92014MR1195559
- [17] 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). Zbl0657.60069MR950166
- [18] N. Dunford and J.T. Schwartz, Linear Operators. Part I. Wiley Classics Library, John Wiley & Sons Inc., New York (1988). Zbl0635.47001MR1009162
- [19] T. Erhardsson, Refined distributional approximations for the uncovered set in the Johnson-Mehl model. Stoch. Proc. Appl.96 (2001) 243–259. Zbl1060.60008MR1865357
- [20] H.J. Frost and C.V. Thompson, The effect of nucleation conditions on the topology and geometry of two-dimensional grain structures. Acta Metallurgica35 (1987) 529–540.
- [21] 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. Zbl0521.60022MR707513
- [22] J. Herrick, S. Jun, J. Bechhoefer and A. Bensimon, Kinetic model of DNA replication in eukaryotic organisms. J. Mol. Biol.320 (2002) 741–750.
- [23] F. Hiai and H. Umegaki, Integrals, conditional expectations, and martingales of multivalued functions. J. Multivariate Anal.7 (1977) 149–182. Zbl0368.60006MR507504
- [24] C.J. Himmelberg, Measurable relations. Fund. Math.87 (1975) 53–72. Zbl0296.28003MR367142
- [25] 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). MR2039695
- [26] G. Matheron, Random Sets and Integral Geometry, John Wiley & Sons, New York-London-Sydney (1975). Zbl0321.60009MR385969
- [27] 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.
- [28] I.S. Molchanov, Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, Chichester (1997). Zbl0878.62068MR1406528
- [29] I.S. Molchanov and S.N. Chiu, Smoothing techniques and estimation methods for nonstationary Boolean models with applications to coverage processes. Biometrika87 (2000) 265–283. Zbl1066.62523MR1782478
- [30] J. Møller, Random Johnson-Mehl tessellations. Adv. Appl. Prob.24 (1992) 814–844. Zbl0768.60014MR1188954
- [31] J. Møller, Generation of Johnson-Mehl crystals and comparative analysis of models for random nucleation. Adv. Appl. Prob.27 (1995) 367–383. Zbl0826.60007MR1334819
- [32] J. Møller and M. Sørensen, 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. Zbl0790.62090MR1267040
- [33] H. Rådström, An embedding theorem for spaces of convex sets. Proc. Am. Math. Soc.3 (1952) 165–169. Zbl0046.33304
- [34] J. Serra, Image Analysis and Mathematical Morphology. Academic Press Inc., London (1984). Zbl0565.92001MR753649
- [35] L. Shoumei and R. Aihong, Representation theorems, set-valued and fuzzy set-valued Ito integral. Fuzzy Set. Syst.158 (2007) 949–962. Zbl1119.60039MR2321701
- [36] D. Stoyan, W.S. Kendall and J. Mecke, Stochastic Geometry and its Applications. 2nd edition, John Wiley & Sons Ltd., Chichester (1995). Zbl1155.60001MR895588

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.