Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Coalescing systems of non-Brownian particles
Download PDF
Download PDF
  • Published: 10 April 2012

Coalescing systems of non-Brownian particles

  • Steven N. Evans1,
  • Ben Morris2 &
  • Arnab Sen1Ā 

Probability Theory and Related Fields volumeĀ 156,Ā pages 307–342 (2013)Cite this article

  • 229 Accesses

  • 4 Citations

  • Metrics details

Abstract

A well-known result of Arratia shows that one can make rigorous the notion of starting an independent Brownian motion at every point of an arbitrary closed subset of the real line and then building a set-valued process by requiring particles to coalesce when they collide. Arratia noted that the value of this process will be almost surely a locally finite set at all positive times, and a finite setĀ almost surely if the initial value is compact: the key to both of these facts is the observation that, because of the topology of the real line and the continuity of Brownian sample paths, at the time when two particles collide one or the other of them must have already collided with each particle that was initially between them. We investigate whether such instantaneous coalescence still occurs for coalescing systems of particles where either the state space of the individual particles is not locally homeomorphic to an interval or the sample paths of the individual particles are discontinuous. We give a quite general criterion for a coalescing system of particles on a compact state space to coalesce to a finite set at all positive times almost surely and show that there is almost sure instantaneous coalescence to a locally finite set for systems of Brownian motions on the Sierpinski gasket and stable processes on the real line with stable index greater than one.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Arratia, R.: Coalescing brownian motions on the line. Ph.D. Thesis (1979)

  2. Arratia, R.: Coalescing brownian motions and the voter model on \({\mathbb Z}\). Unpublished partial manuscript (1981)

  3. Barlow, M.T.: Diffusions on fractals. Lectures on Probability Theory and Statistics (Saint-Flour, 1995). Lecture Notes in Math., vol. 1690, pp. 1–121. Springer, Berlin (1998)

  4. Bertoin J.: LƩvy processes. Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996)

    Google ScholarĀ 

  5. Barlow M.T., Perkins E.A.: Brownian motion on the Sierpinski gasket. Probab. Theory Relat. Fields 79(4), 543–623 (1988)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  6. Chen Z.-Q., Kumagai T.: Heat kernel estimates for stable-like processes on d-sets. Stoch. Process. Appl. 108(1), 27–62 (2003)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  7. Donnelly P., Evans S.N., Fleischmann K., Kurtz T.G., Zhou X.: Continuum-sites stepping-stone models, coalescing exchangeable partitions and random trees. Ann. Probab. 28(3), 1063–1110 (2000)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  8. Doob, J.L.: Classical potential theory and its probabilistic counterpart. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1984 edition

  9. Evans S.N., Fleischmann K.: Cluster formation in a stepping-stone model with continuous, hierarchically structured sites. Ann. Probab. 24(4), 1926–1952 (1996)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  10. Evans S.N.: Coalescing Markov labelled partitions and a continuous sites genetics model with infinitely many types. Ann. Inst. H. PoincarĆ© Probab. Statist. 33(3), 339–358 (1997)

    ArticleĀ  MATHĀ  Google ScholarĀ 

  11. Fontes L.R.G., Isopi M., Newman C.M., Ravishankar K.: The Brownian web: characterization and convergence. Ann. Probab. 32(4), 2857–2883 (2004)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  12. Hambly B.M., Kumagai T.: Diffusion processes on fractal fields: heat kernel estimates and large deviations. Probab. Theory Relat. Fields 127(3), 305–352 (2003)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  13. Hobson, T., Tribe, R.: On the duality between coalescing Brownian particles and the heat equation driven by Fisher-Wright noise. Electron. Commun. Probab. 10, 136–145 (2005)

    Google ScholarĀ 

  14. Howitt C., Warren J.: Dynamics for the Brownian web and the erosion flow. Stoch. Process. Appl. 119(6), 2028–2051 (2009)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  15. Klenke A.: Different clustering regimes in systems of hierarchically interacting diffusions. Ann. Probab. 24(2), 660–697 (1996)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  16. Kumagai T., Sturm K.-T.: Construction of diffusion processes on fractals, d-sets, and general metric measure spaces. J. Math. Kyoto Univ. 45(2), 307–327 (2005)

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  17. LindstrĆøm T.: Brownian motion on nested fractals. Mem. Am. Math. Soc. 83(420), iv+128 (1990)

    Google ScholarĀ 

  18. Le Jan Y., Raimond O.: Flows, coalescence and noise. Ann. Probab. 32(2), 1247–1315 (2004)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  19. Munasinghe R., Rajesh R., Tribe R., Zaboronski O.: Multi-scaling of the n-point density function for coalescing Brownian motions. Commun. Math. Phys. 268(3), 717–725 (2006)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  20. Soucaliuc F., Tóth B., Werner W.: Reflection and coalescence between independent one-dimensional Brownian paths. Ann. Inst. H. PoincarĆ© Probab. Statist. 36(4), 509–545 (2000)

    ArticleĀ  MATHĀ  Google ScholarĀ 

  21. Soucaliuc, F., Werner, W.: A note on reflecting Brownian motions. Electron. Commun. Probab. 7, 117–122 (2002)

    Google ScholarĀ 

  22. Tsirelson, B.: Scaling limit, noise, stability. In: Lectures on Probability Theory and Statistics. Lecture Notes in Math., vol. 1840, pp. 1–106. Springer, Berlin (2004)

  23. Tóth B., Werner W.: The true self-repelling motion. Probab. Theory Relat. Fields 111(3), 375–452 (1998)

    ArticleĀ  MATHĀ  Google ScholarĀ 

  24. Xiong J., Zhou X.: On the duality between coalescing Brownian motions. Can. J. Math. 57(1), 204–224 (2005)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  25. Zhou, X.: Clustering behavior of a continuous-sites stepping-stone model with Brownian migration. Electron. J. Probab. 8(11), p. 15 (2003)

    Google ScholarĀ 

  26. Zhou X.: Stepping-stone model with circular Brownian migration. Can. Math. Bull. 51(1), 146–160 (2008)

    ArticleĀ  MATHĀ  Google ScholarĀ 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics #3860, University of California at Berkeley, 367 Evans Hall, Berkeley, CA, 94720-3860, USA

    Steven N. EvansĀ &Ā Arnab Sen

  2. Department of Mathematics, University of California at Davis, Mathematical Sciences Building, One Shields Avenue, Davis, CA, 95616, USA

    Ben Morris

Authors
  1. Steven N. Evans
    View author publications

    You can also search for this author in PubMedĀ Google Scholar

  2. Ben Morris
    View author publications

    You can also search for this author in PubMedĀ Google Scholar

  3. Arnab Sen
    View author publications

    You can also search for this author in PubMedĀ Google Scholar

Corresponding author

Correspondence to Steven N. Evans.

Additional information

S. N. Evans supported in part by NSF grants DMS-0405778 and DMS-0907630 and B. Morris supported in part by NSF grant DMS-0707144.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Evans, S.N., Morris, B. & Sen, A. Coalescing systems of non-Brownian particles. Probab. Theory Relat. Fields 156, 307–342 (2013). https://doi.org/10.1007/s00440-012-0429-0

Download citation

  • Received: 30 November 2009

  • Revised: 17 March 2012

  • Published: 10 April 2012

  • Issue Date: June 2013

  • DOI: https://doi.org/10.1007/s00440-012-0429-0

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Stepping stone model
  • Brownian web
  • Fractal
  • Hitting time
  • Coalescing particle system

Mathematics Subject Classification

  • 60G17
  • 60G52
  • 60J60
  • 60K35
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature