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

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

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 2, page 537-574
  • ISSN: 0246-0203

Abstract

top
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.

How to cite

top

Newman, C. M., Ravishankar, K., and Schertzer, E.. "Marking (1, 2) points of the brownian web and applications." Annales de l'I.H.P. Probabilités et statistiques 46.2 (2010): 537-574. <http://eudml.org/doc/239784>.

@article{Newman2010,
abstract = {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.},
author = {Newman, C. M., Ravishankar, K., Schertzer, E.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {brownian web; brownian net; dynamical brownian web; coalescing random walks; poissonian marking; nucleation on boundaries; sticky brownian motion; Brownian web; Brownian net; dynamical Brownian web; Poissonian marking; sticky Brownian motion},
language = {eng},
number = {2},
pages = {537-574},
publisher = {Gauthier-Villars},
title = {Marking (1, 2) points of the brownian web and applications},
url = {http://eudml.org/doc/239784},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Newman, C. M.
AU - Ravishankar, K.
AU - Schertzer, E.
TI - Marking (1, 2) points of the brownian web and applications
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 2
SP - 537
EP - 574
AB - 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.
LA - eng
KW - brownian web; brownian net; dynamical brownian web; coalescing random walks; poissonian marking; nucleation on boundaries; sticky brownian motion; Brownian web; Brownian net; dynamical Brownian web; Poissonian marking; sticky Brownian motion
UR - http://eudml.org/doc/239784
ER -

References

top
  1. [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. [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. Zbl1021.60055MR1959784
  3. [3] A. Borodin and P. Salminen. Handbook of Brownian Motion—Facts and Formulae, 2nd edition. Birkhäuser, Basel, 2002. Zbl0859.60001MR1912205
  4. [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. Zbl1119.82027MR2282484
  5. [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. Zbl1109.60326MR2284886
  6. [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. Zbl1179.82075MR2556735
  7. [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. Zbl1105.60075MR2094432
  8. [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. Zbl1087.60072MR2196970
  9. [9] L. R. G. Fontes, C. M. Newman, K. Ravishankar and E. Schertzer. The dynamical discrete web, 2007. Available at arXiv:0704.2706. Zbl1172.60334
  10. [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. Zbl1172.60334MR2554030
  11. [11] C. Garban. Processus SLE et sensibilité aux perturbations de la percolation plane critique. Doctoral thesis, Univ. Paris Sud, Paris, 2008. 
  12. [12] C. Garban, G. Pete and O. Schramm. Private communication, 2007. 
  13. [13] C. Howitt and J. Warren. Consistent families of Brownian motions and stochastic flows of kernels. Ann. Probab. 37 (2009) 1237–1272. Zbl1203.60123MR2546745
  14. [14] C. Howitt and J. Warren. Dynamics for the Brownian web and the erosion flow. Stochastic Process. Appl. 119 (2009) 2028–2051. Zbl1163.60046MR2519355
  15. [15] I. Karatzas and E. Shreve. Brownian Motion and Stochastic Calculus. Springer, New York, 1991. Zbl0734.60060MR1121940
  16. [16] C. M. Newman, K. Ravishankar and E. Schertzer. Scaling limit of the one-dimensional stochastic Potts model, 2009. In preparation. 
  17. [17] P. Nolin. Near-critical percolation in two dimensions. Electron. J. Probab. 13 (2008) 1562–1623. Zbl1189.60182MR2438816
  18. [18] P. Nolin and W. Werner. Asymmetry of near-critical percolation interfaces. J. Amer. Math. Soc. 22 (2009) 797–819. Zbl1218.60088MR2505301
  19. [19] E. Perkins. The exact Hausdorff measure of the level sets of Brownian motions. Z. Wahrsch. Verw. Gebiete 58 (1981) 373–388. Zbl0458.60076MR639146
  20. [20] E. Schertzer, R. Sun and J.M. Swart. Special points of the Brownian net. Electron. J. Probab. 14 (2009) 805–864. Zbl1187.82081MR2497454
  21. [21] E. Schertzer, R. Sun and J.M. Swart. Stochastic flows in the Brownian web and net, 2009. In preparation. Zbl1303.82028
  22. [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. Zbl0968.60072MR1785393
  23. [23] R. Sun and J. M. Swart. The Brownian net. Ann. Probab. 36 (2008) 1153–1208. Zbl1143.82020MR2408586
  24. [24] B. Tóth and W. Werner. The true self-repelling motion. Probab. Theory Related Fields 111 (1998) 375–452. Zbl0912.60056MR1640799
  25. [25] B. Tsirelson. Scaling limit, noise, stability. In Lectures in Probability Theory and Statistics 1–106. Lecture Notes in Math. 1840. Springer, Berlin, 2004. Zbl1056.60009MR2079671
  26. [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. Zbl0884.60081MR1478711
  27. [27] J. Warren. The noise made by a Poisson snake. Electron. J. Probab. 7 (2002), no. 21. Zbl1007.60052MR1943894

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.