The Annals of Probability

The Brownian net

Rongfeng Sun and Jan M. Swart

Full-text: Open access

Abstract

The (standard) Brownian web is a collection of coalescing one- dimensional Brownian motions, starting from each point in space and time. It arises as the diffusive scaling limit of a collection of coalescing random walks. We show that it is possible to obtain a nontrivial limiting object if the random walks in addition branch with a small probability. We call the limiting object the Brownian net, and study some of its elementary properties.

Article information

Source
Ann. Probab., Volume 36, Number 3 (2008), 1153-1208.

Dates
First available in Project Euclid: 9 April 2008

Permanent link to this document
https://projecteuclid.org/euclid.aop/1207749093

Digital Object Identifier
doi:10.1214/07-AOP357

Mathematical Reviews number (MathSciNet)
MR2408586

Zentralblatt MATH identifier
1143.82020

Subjects
Primary: 82C21: Dynamic continuum models (systems of particles, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60F17: Functional limit theorems; invariance principles 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Brownian net Brownian web branching-coalescing random walks branching-coalescing point set

Citation

Sun, Rongfeng; Swart, Jan M. The Brownian net. Ann. Probab. 36 (2008), no. 3, 1153--1208. doi:10.1214/07-AOP357. https://projecteuclid.org/euclid.aop/1207749093


Export citation

References

  • [1] Arratia, R. (1979). Coalescing Brownian motions on the line. Ph.D. thesis, Univ. Wisconsin, Madison.
  • [2] Arratia, R. Coalescing Brownian motions and the voter model on ℤ. Unpublished partial manuscript. Available from rarratia@math.usc.edu.
  • [3] Athreya, S. R. and Swart, J. M. (2005). Branching-coalescing particle systems. Probab. Theory Related Fields 131 376–414.
  • [4] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. J. Wiley, New York.
  • [5] Ding, W., Durrett, R. and Liggett, T. M. (1990). Ergodicity of reversible reaction diffusion processes. Probab. Theory Related Fields 85 13–26.
  • [6] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. Wiley, New York.
  • [7] Fontes, L. R. G., Isopi, M., Newman, C. M. and Ravishankar, K. (2002). The Brownian web. Proc. Natl. Acad. Sci. 99 15888–15893.
  • [8] Fontes, L. R. G., Isopi, M., Newman, C. M. and Ravishankar, K. (2004). The Brownian web: Characterization and convergence. Ann. Probab. 32 2857–2883.
  • [9] Fontes, L. R. G., Isopi, M., Newman, C. M. and Ravishankar, K. (2006). Coarsening, nucleation, and the marked Brownian web. Ann. Inst. H. Poincaré Probab. Statist. 42 37–60.
  • [10] Howitt, C. and Warren, J. Consistent families of Brownian motions and stochastic flows of kernels. ArXiv: math.PR/0611292.
  • [11] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
  • [12] Newman, C. M., Ravishankar, K. and Schertzer, E. Marking (1, 2) points of the Brownian web and applications. In preparation.
  • [13] Schlögl, F. (1972). Chemical reaction models and non-equilibrium phase transitions. Z. Phys. 253 147–161.
  • [14] Schertzer, E., Sun, R. and Swart, J. M. Special points of the Brownian net. In preparation.
  • [15] Soucaliuc, F., Tóth, B. and Werner, W. (2000). Reflection and coalescence between one-dimensional Brownian paths. Ann. Inst. H. Poincaré Probab. Statist. 36 509–536.
  • [16] Sudbury, A. (1999). Hunting submartingales in the jumping voter model and the biased annihilating branching process. Adv. in Appl. Probab. 31 839–854.
  • [17] Tóth, B. and Werner, W. (1998). The true self-repelling motion. Probab. Theory Related Fields 111 375–452.
  • [18] Warren, J. (2002). The noise made by a Poisson snake. Electron. J. Probab. 7 1–21.