Convergence to the brownian Web for a generalization of the drainage network model
Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 3, pp. 899-919.

Nous introduisons un système de marches aléatoires coalescentes unidimensionnelles, avec des sauts à longue portée. Ce système autorise des chemins qui se croisent, et qui sont dépendants, même avant leur coalescence. Après une renormalisation diffusive, nous montrons que ce système converge en loi vers le réseau brownien.

We introduce a system of one-dimensional coalescing nonsimple random walks with long range jumps allowing paths that can cross each other and are dependent even before coalescence. We show that under diffusive scaling this system converges in distribution to the Brownian Web.

DOI : 10.1214/13-AIHP544
Classification : 60K35
Mots-clés : brownian web, coalescing brownian motions, coalescing random walks, drainage network, scaling limit, invariance principle, interacting particle systems 
@article{AIHPB_2014__50_3_899_0,
     author = {Coletti, Cristian and Valle, Glauco},
     title = {Convergence to the brownian {Web} for a generalization of the drainage network model},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {899--919},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {3},
     year = {2014},
     doi = {10.1214/13-AIHP544},
     mrnumber = {3224293},
     zbl = {1296.60080},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/13-AIHP544/}
}
TY  - JOUR
AU  - Coletti, Cristian
AU  - Valle, Glauco
TI  - Convergence to the brownian Web for a generalization of the drainage network model
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2014
SP  - 899
EP  - 919
VL  - 50
IS  - 3
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/13-AIHP544/
DO  - 10.1214/13-AIHP544
LA  - en
ID  - AIHPB_2014__50_3_899_0
ER  - 
%0 Journal Article
%A Coletti, Cristian
%A Valle, Glauco
%T Convergence to the brownian Web for a generalization of the drainage network model
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2014
%P 899-919
%V 50
%N 3
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/13-AIHP544/
%R 10.1214/13-AIHP544
%G en
%F AIHPB_2014__50_3_899_0
Coletti, Cristian; Valle, Glauco. Convergence to the brownian Web for a generalization of the drainage network model. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 3, pp. 899-919. doi : 10.1214/13-AIHP544. http://archive.numdam.org/articles/10.1214/13-AIHP544/

[1] R. Arratia. Coalescing Brownian motions and the voter model on ℤ. Unpublished partial manuscript, 1981.

[2] R. Arratia. Limiting point processes for rescalings of coalescing and annihilating random walks on ℤ d . Ann. Probab. 9 (1981) 909-936. | MR | Zbl

[3] S. Belhaouari, T. Mountford, R. Sun and G. Valle. Convergence results and sharp estimates for the voter model interfaces. Electron. J. Probab. 11 (2006) 768-801. | MR | Zbl

[4] C. Coletti, L. R. Fontes and E. Dias. Scaling limit for a drainage network model. J. Appl. Probab. 46 (2009) 1184-1197. | MR | Zbl

[5] R. Durrett. Probability: Theory and Examples, 2nd edition. Duxbury Press, Belmont, CA, 1996. | MR | Zbl

[6] P. A. Ferrari, L. R. G. Fontes and X. Y. Wu. Two dimensional Poisson trees converge to the Brownian web. Ann. Inst. Henri Poincaré Probab. Stat. 41 (2005) 851-858. | Numdam | MR | Zbl

[7] L. R. G. Fontes, M. Isopi, C. M. Newman and K. Ravishankar. The Brownian web. Proc. Natl. Acad. Sci. USA 99 (2002) 15888-15893. | MR | Zbl

[8] L. R. G. Fontes, M. Isopi, C. M. Newman and K. Ravishankar. The Brownian web: Characterization and convergence. Ann. Probab. 32 (2004) 2857-2883. | MR | Zbl

[9] L. R. G. Fontes, M. Isopi, C. M. Newman and K. Ravishankar. Coarsening, nucleation, and the marked Brownian web. Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 37-60. | Numdam | MR | Zbl

[10] S. Gangopadhyay, R. Roy and A. Sarkar. Random oriented trees: A model of drainage networks. Ann. App. Probab. 14 (2004) 1242-1266. | MR | Zbl

[11] C. M. Newman, K. Ravishankar and R. Sun. Convergence of coalescing nonsimple random walks to the Brownian web. Electron. J. Probab. 10 (2005) 21-60. | EuDML | MR | Zbl

[12] R. Sun. Convergence of coalescing nonsimple random walks to the Brownian web. Ph.D. thesis, Courant Institute of Mathematical Sciences, New York Univ., 2005. Available at arXiv:math/0501141. | MR | Zbl

[13] A. Sarkar and R. Sun. Brownian web in the scaling limit of supercritical oriented percolation in dimension 1+1. Electron. J. Probab. 18 (2013) Article 21. | MR | Zbl

[14] R. Sun and J. M. Swart. The Brownian net. Ann. Probab. 36 (2008) 1153-1208. | MR | Zbl

[15] B. Tóth and W. Werner. The true self-repelling motion. Probab. Theory Related Fields 111 (1998) 375-452. | MR | Zbl

Cité par Sources :