Marking (1, 2) points of the brownian web and applications

Newman, C. M.; Ravishankar, K.; Schertzer, E.

Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 2, pp. 537-574.

Le réseau brownien (BW) construit à partir des travaux de Arratia, de Tòth et de Werner est une collection aléatoire de chemins (avec des points de départ déterminés) dans un espace deux-dimensionnel (une dimension en temps et une autre en espace), qui est la limite d'échelle d'un réseau discret (DW) de marches aléatoires coalescentes. Récemment, deux extensions du BW ont été introduites: le filet Brownien (BN), construit par Sun et Swart, et le réseau Brownien dynamique (DyBW), proposé par Howitt et Warren. Ces deux objets sont (ou devraient être) la limite d'échelle de deux extensions naturelles du réseau discret - le filet discret (DN) et le réseau dynamique discret (DyDW). Le DN et le DyDW sont obtenus par une modification de la configuration des “flèches” droites ou gauches qui composent le réseau discret. Pour le DN, un mécanisme de ramification est introduit (en permettant des flèches droites et gauches simultanément) alors que pour le DyDW, la direction des flèches est modifiée (de droite à gauche et vice-versa). Dans cet article, nous montrons qu'il existe une structure géométrique analogue dans le cas continu. Plus précisément, la direction des flèches dans le cas discret est remplacée par la direction des points (1, 2) du réseau Brownien (en un point (1, 2) se trouvent un chemin entrant et deux chemins sortants, l'un d'eux étant la continuation du chemin entrant). Nous montrons que les ramifications et changements de direction peuvent être introduits dans le cas continu par un marquage de type Poisson des points (1, 2). Par l'intermédiaire de ce marquage, nous donnons une construction complète du DyBW et une construction alternative à celle de Sun et Swart du BN.

The brownian web (BW), which developed from the work of Arratia and then Tóth and Werner, is a random collection of paths (with specified starting points) in one plus one dimensional space-time that arises as the scaling limit of the discrete web (DW) of coalescing simple random walks. Two recently introduced extensions of the BW, the brownian net (BN) constructed by Sun and Swart, and the dynamical brownian web (DyBW) proposed by Howitt and Warren, are (or should be) scaling limits of corresponding discrete extensions of the DW - the discrete net (DN) and the dynamical discrete web (DyDW). These discrete extensions have a natural geometric structure in which the underlying Bernoulli left or right “arrow” structure of the DW is extended by means of branching (i.e., allowing left and right simultaneously) to construct the DN or by means of switching (i.e., from left to right and vice-versa) to construct the DyDW. In this paper we show that there is a similar structure in the continuum where arrow direction is replaced by the left or right parity of the (1, 2) space-time points of the BW (points with one incoming path from the past and two outgoing paths to the future, only one of which is a continuation of the incoming path). We then provide a complete construction of the DyBW and an alternate construction of the BN to that of Sun and Swart by proving that the switching or branching can be implemented by a poissonian marking of the (1, 2) points.

[1] R. Arratia. Coalescing Brownian motions on ℝ and the voter model on ℤ. Unpublished partial manuscript, 1981. Available at http://almaak.usc.edu/ rarratia/.

[2] I. Benjamini, O. Häggstrom, Y. Peres and J. E. Steif. Which properties of a random sequence are dynamically sensitive? Ann. Probab. 31 (2003) 1-34. | MR | Zbl

[3] A. Borodin and P. Salminen. Handbook of Brownian Motion-Facts and Formulae, 2nd edition. Birkhäuser, Basel, 2002. | MR | Zbl

[4] F. Camia, L. R. G. Fontes and C.M. Newman. The scaling limit geometry of near-critical 2D percolation. J. Statist. Phys. 125 (2006) 1159-1175. | MR | Zbl

[5] F. Camia, L. R. G. Fontes and C. M. Newman. Two-dimensional scaling limits via marked nonsimple loops. Bull. Braz. Math. Soc. (N. S.) 37 (2006) 537-559. | MR | Zbl

[6] F. Camia, M. Joosten and R. Meester. Geometric properties of two-dimensional near-critical percolation. Revised version submitted in 2009 as Trivial, critical and near-critical scaling limits of two-dimensional percolation. J. Statist. Phys. To appear, 2010. Available at arXiv:0803.3785. | MR | Zbl

[7] 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

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

[9] L. R. G. Fontes, C. M. Newman, K. Ravishankar and E. Schertzer. The dynamical discrete web, 2007. Available at arXiv:0704.2706.

[10] L. R. G. Fontes, C. M. Newman, K. Ravishankar and E. Schertzer. Exceptional times for the dynamical discrete web. Stochastic Process. Appl. 119 (2009) 2832-2858. | MR | Zbl

[11] C. Garban. Processus SLE et sensibilité aux perturbations de la percolation plane critique. Doctoral thesis, Univ. Paris Sud, Paris, 2008.

[12] C. Garban, G. Pete and O. Schramm. Private communication, 2007.

[13] C. Howitt and J. Warren. Consistent families of Brownian motions and stochastic flows of kernels. Ann. Probab. 37 (2009) 1237-1272. | MR | Zbl

[14] C. Howitt and J. Warren. Dynamics for the Brownian web and the erosion flow. Stochastic Process. Appl. 119 (2009) 2028-2051. | MR | Zbl

[15] I. Karatzas and E. Shreve. Brownian Motion and Stochastic Calculus. Springer, New York, 1991. | MR | Zbl

[16] C. M. Newman, K. Ravishankar and E. Schertzer. Scaling limit of the one-dimensional stochastic Potts model, 2009. In preparation.

[17] P. Nolin. Near-critical percolation in two dimensions. Electron. J. Probab. 13 (2008) 1562-1623. | MR | Zbl

[18] P. Nolin and W. Werner. Asymmetry of near-critical percolation interfaces. J. Amer. Math. Soc. 22 (2009) 797-819. | MR | Zbl

[19] E. Perkins. The exact Hausdorff measure of the level sets of Brownian motions. Z. Wahrsch. Verw. Gebiete 58 (1981) 373-388. | MR | Zbl

[20] E. Schertzer, R. Sun and J.M. Swart. Special points of the Brownian net. Electron. J. Probab. 14 (2009) 805-864. | MR | Zbl

[21] E. Schertzer, R. Sun and J.M. Swart. Stochastic flows in the Brownian web and net, 2009. In preparation.

[22] F. Soucaliuc, B. Tóth and W. Werner. Reflection and coalescence between independent one-dimensional Brownian paths. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) 509-545. | Numdam | MR | Zbl

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

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

[25] B. Tsirelson. Scaling limit, noise, stability. In Lectures in Probability Theory and Statistics 1-106. Lecture Notes in Math. 1840. Springer, Berlin, 2004. | MR | Zbl

[26] J. Warren. Branching processes, the Ray-Knight theorem, and sticky Brownian motion. In Séminaire de Probabilité de Strasbourg 31 1-15. Lecture Notes in Math. 1655. Springer, Berlin, 1997. | Numdam | MR | Zbl

[27] J. Warren. The noise made by a Poisson snake. Electron. J. Probab. 7 (2002), no. 21. | MR | Zbl