Special points of the Brownian net

Emmanuel Schertzer (Columbia University)
Rongfeng Sun (National University of Singapore)
Jan M. Swart (Institute of Information Theory and Automation of the ASCR (UTIA))

Abstract


The Brownian net, which has recently been introduced by Sun and Swart [16], and independently by Newman, Ravishankar and Schertzer [13], generalizes the Brownian web by allowing branching. In this paper, we study the structure of the Brownian net in more detail. In particular, we give an almost sure classification of each point in $\mathbb{R}^2$ according to the configuration of the Brownian net paths entering and leaving the point. Along the way, we establish various other structural properties of the Brownian net.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 805-864

Publication Date: April 19, 2009

DOI: 10.1214/EJP.v14-641

References

  1. R. Arratia. Coalescing Brownian motions on the line. Ph.D. Thesis, University of Wisconsin, Madison, 1979. Math. Review number not available.
  2. R. Arratia. Coalescing Brownian motions and the voter model on Z. Unpublished partial manuscript, 1981. Available from rarratia@math.usc.edu. Math. Review number not available.
  3. J. Bertoin. Lévy processes. Cambridge Tracts in Mathematics. 121. Cambridge Univ. Press, 1996. MR1406564
  4. P. Billingsley. Convergence of probability measures, 2nd edition. John Wiley & Sons, 1999. MR1700749 Zbl 0944.60003
  5. L.R.G. Fontes, M. Isopi, C.M. Newman, K. Ravishankar. The Brownian web: characterization and convergence. Ann. Probab. 32(4) (2004), 2857-2883. MR2094432 Zbl 1105.60075
  6. L.R.G. Fontes, M. Isopi, C.M. Newman, K. Ravishankar. Coarsening, nucleation, and the marked Brownian web. Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), 37-60. MR2196970 Zbl 1087.60072
  7. L.R.G. Fontes, C.M. Newman, K. Ravishankar, E. Schertzer. The dynamical discrete web. (2007) ArXiv: 0704.2706. Math. Review number not available.
  8. O. Häggström. Dynamical percolation: early results and open problems. Microsurveys in discrete probability, 59-74, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 41 (1998). MR1630409 Zbl 0906.60081
  9. C. Howitt and J. Warren. Consistent families of Brownian motions and stochastic flows of kernels. (2006) To appear in Ann. Probab. ArXiv: math.PR/0611292. Math. Review number not available.
  10. C. Howitt and J. Warren, Dynamics for the Brownian web and the erosion flow. (2007) ArXiv: math.PR/0702542. Math. Review number not available.
  11. I. Karatzas, S.E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition, Springer-Verlag, New York, 1991. MR1121940 Zbl 0734.60060
  12. C.M. Newman, K. Ravishankar, R. Sun. Convergence of coalescing nonsimple random walks to the Brownian web. Electron. J. Prob. 10 (2005), 21-60. MR2120239 Zbl 1067.60099
  13. C.M. Newman, K. Ravishankar, E. Schertzer. Marking (1,2) points of the Brownian web and applications. (2008) ArXiv:0806.0158v1. Math. Review number not available.
  14. C.M. Newman, K. Ravishankar, E. Schertzer. The scaling limit of the one-dimensional stochastic Potts model. In preparation. Math. Review number not available.
  15. L.C.G. Rogers, D. Williams. Diffusions, Markov processes, and martingales, Vol. 2, 2nd edition. Cambridge University Press, Cambridge, 2000. Math. Review number: no unique match found. Zbl 0977.60005
  16. R. Sun and J.M. Swart. The Brownian net. Ann. Probab. 36(3) (2008), 1153-1208. MR2408586 Zbl 1143.82020
  17. E. Schertzer, R. Sun, J.M. Swart. Stochastic flows in the Brownian web and net. In preparation. Math. Review number not available.
  18. F. Soucaliuc, B. Tóth, W. Werner. Reflection and coalescence between one-dimensional Brownian paths. Ann. Inst. Henri Poincaré Probab. Statist. 36(4) (2000), 509-536. MR1785393 Zbl 0968.60072
  19. B. Tóth and W. Werner. The true self-repelling motion. Probab. Theory Related Fields 111(3) (1998), 375-452. MR1640799 Zbl 0912.60056
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.