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

Abstract

top
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.

How to cite

top

Aletti, 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. [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. [2] G. Aletti and D. Saada, Survival analysis in Johnson-Mehl tessellation. Stat. Infer. Stoch. Process.11 (2008) 55–76. Zbl1148.62080MR2357553
  3. [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. [4] J. Aubin and H. Frankowska, Set-valued Analysis. Birkhäuser, Boston Inc. (1990). Zbl1168.49014MR1048347
  5. [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. [6] M. Burger, Growth fronts of first-order Hamilton-Jacobi equations. SFB Report 02-8, University Linz, Linz, Austria (2002). 
  7. [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. [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. [9] V. Capasso (Ed.) Mathematical Modelling for Polymer Processing. Polymerization, Crystallization, Manufacturing. Mathematics in Industry 2, Springer-Verlag, Berlin (2003). Zbl1004.00015MR1964653
  10. [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. [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. [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. [13] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions. Lecture Notes Math. 580, Springer-Verlag, Berlin (1977). Zbl0346.46038MR467310
  14. [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. [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. [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. [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. [18] N. Dunford and J.T. Schwartz, Linear Operators. Part I. Wiley Classics Library, John Wiley & Sons Inc., New York (1988). Zbl0635.47001MR1009162
  19. [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. [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. [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. [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. [23] F. Hiai and H. Umegaki, Integrals, conditional expectations, and martingales of multivalued functions. J. Multivariate Anal.7 (1977) 149–182. Zbl0368.60006MR507504
  24. [24] C.J. Himmelberg, Measurable relations. Fund. Math.87 (1975) 53–72. Zbl0296.28003MR367142
  25. [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. [26] G. Matheron, Random Sets and Integral Geometry, John Wiley & Sons, New York-London-Sydney (1975). Zbl0321.60009MR385969
  27. [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. [28] I.S. Molchanov, Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, Chichester (1997). Zbl0878.62068MR1406528
  29. [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. [30] J. Møller, Random Johnson-Mehl tessellations. Adv. Appl. Prob.24 (1992) 814–844. Zbl0768.60014MR1188954
  31. [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. [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. [33] H. Rådström, An embedding theorem for spaces of convex sets. Proc. Am. Math. Soc.3 (1952) 165–169. Zbl0046.33304
  34. [34] J. Serra, Image Analysis and Mathematical Morphology. Academic Press Inc., London (1984). Zbl0565.92001MR753649
  35. [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. [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 ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.