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Stochastic p.d.e.'s arising from the long range contact and long range voter processes
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  • Published: December 1995

Stochastic p.d.e.'s arising from the long range contact and long range voter processes

  • C. Müller1 &
  • R. Tribe2 

Probability Theory and Related Fields volume 102, pages 519–545 (1995)Cite this article

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  • 57 Citations

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Summary

A long range contact process and a long range voter process are scaled so that the distance between sites decreases and the number of neighbors of each site increases. The approximate densities of occupied sites, under suitable tine scaling, converge to continuous space time densities which solve stochastic p.d.e.'s. For the contact process the limiting equation is the Kolmogorov-Petrovskii-Piscuinov equation driven by branching white noise. For the voter process the limiting equation is the heat equation driven by Fisher-Wright white noise.

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References

  1. Bhattacharya, R.N., Rao, R.R.: Normal approximation and asymptotic expansions. New York: Wiley 1976

    Google Scholar 

  2. Burkholder, D.L.: Distribution function inequalities for martingales. Ann. Probab.1, 19–42, 1973

    Google Scholar 

  3. Dawson, D.A.: Measure valued Markov processes, Ecole d'ete de probabilites de saint Flour 1991 (Lecture Notes in Mathematics) Berlin: Springer

    Google Scholar 

  4. Durrett, R.: Lecture notes on particle systems and percolation. Wadsworth and Brooks & Cole, 1988

  5. Durrett, R.: Probability: Theory and examples. Wadsworth and Brooks & Cole, 1991

  6. Ethier, S.N., Kurtz, T.G.: Markov processes, characterization and covergence. New York: Wiley 1986

    Google Scholar 

  7. Evans, S.N., Perkins, E.A.: Measure-valued diffusions with singular interactions (preprint) 1993

  8. Mueller, C., Sowers, R.: Random travelling waves for the K.P.P. equation with noise, J. Fun. Anal. 1995 2. p 439–498

    Google Scholar 

  9. Mueller, C., Tribe, R.: A phase transition for a stochastic P.D.E. arising from the long range contact process. Probab. Theory Related Fields, 1994 (to appear)

  10. Perkins, E.A.: On a problem of Durrett. Handwritten manuscript, 1988

  11. Rogers, L.C.G., Williams, D.: Diffusions, Markov processes, and marrtingles Vol. 2. New York: Wiley, 1987

    Google Scholar 

  12. Shiga, T.: Stepping stone models in population genetics and population dynamics. Stochastic Process in Physics and Engineering. 1988

  13. Shiga, T.: Two contrastive properties of solutions for our dimensional stochastic partial differential equations Can. J. Math 1993 46. p 415–437

    Google Scholar 

  14. Tribe, R.: Travelling wave solutions for the Kolmogorov equation driven by white noise, Weierstrass Institute for Applied Analysis Preprint No. 73, 1994

  15. Tribe, R.: Large time behavior of interface solutions to a heat equation with Fisher-Wright white noise Probab. Theory Related Fields. 1995 (to appear)

  16. Walsh, J.B.: A introduction to stochastic partial differental equations (Lecture notes in Mathematics, 1180, 265–439) Berlin: Springer 1986

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Rochester, Rochester, USA

    C. Müller

  2. Mathematics Institute, University of Warwick, Warwick, USA

    R. Tribe

Authors
  1. C. Müller
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  2. R. Tribe
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Müller, C., Tribe, R. Stochastic p.d.e.'s arising from the long range contact and long range voter processes. Probab. Th. Rel. Fields 102, 519–545 (1995). https://doi.org/10.1007/BF01198848

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  • Received: 25 May 1994

  • Revised: 30 January 1995

  • Issue Date: December 1995

  • DOI: https://doi.org/10.1007/BF01198848

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Mathematics Subject Classification

  • 60H25
  • 35R60
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