The Annals of Probability

Infinite rate mutually catalytic branching in infinitely many colonies: The longtime behavior

Achim Klenke and Leonid Mytnik

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Abstract

Consider the infinite rate mutually catalytic branching process (IMUB) constructed in [Infinite rate mutually catalytic branching in infinitely many colonies. Construction, characterization and convergence (2008) Preprint] and [Ann. Probab. 38 (2010) 479–497]. For finite initial conditions, we show that only one type survives in the long run if the interaction kernel is recurrent. On the other hand, under a slightly stronger condition than transience, we show that both types can coexist.

Article information

Source
Ann. Probab., Volume 40, Number 1 (2012), 103-129.

Dates
First available in Project Euclid: 3 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1325604998

Digital Object Identifier
doi:10.1214/10-AOP621

Mathematical Reviews number (MathSciNet)
MR2917768

Zentralblatt MATH identifier
1244.60088

Subjects
Primary: 60J65: Brownian motion [See also 58J65] 60G17: Sample path properties 60J55: Local time and additive functionals 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Mutually catalytic branching Lévy noise stochastic differential equations Trotter product coexistence segregation of types

Citation

Klenke, Achim; Mytnik, Leonid. Infinite rate mutually catalytic branching in infinitely many colonies: The longtime behavior. Ann. Probab. 40 (2012), no. 1, 103--129. doi:10.1214/10-AOP621. https://projecteuclid.org/euclid.aop/1325604998


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References

  • [1] Clifford, P. and Sudbury, A. (1973). A model for spatial conflict. Biometrika 60 581–588.
  • [2] Cox, J. T. and Greven, A. (1994). Ergodic theorems for infinite systems of locally interacting diffusions. Ann. Probab. 22 833–853.
  • [3] Cox, J. T., Klenke, A. and Perkins, E. A. (2000). Convergence to equilibrium and linear systems duality. In Stochastic Models (Ottawa, ON, 1998) (L. B. Gorostiza and B. G. Ivanoff, eds.). CMS Conf. Proc. 26 41–66. Amer. Math. Soc., Providence, RI.
  • [4] Dawson, D. A. and Fleischmann, K. (1985). Critical dimension for a model of branching in a random medium. Z. Wahrsch. Verw. Gebiete 70 315–334.
  • [5] Dawson, D. A. and Perkins, E. A. (1998). Long-time behavior and coexistence in a mutually catalytic branching model. Ann. Probab. 26 1088–1138.
  • [6] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [7] Holley, R. and Liggett, T. M. (1981). Generalized potlatch and smoothing processes. Z. Wahrsch. Verw. Gebiete 55 165–195.
  • [8] Holley, R. A. and Liggett, T. M. (1975). Ergodic theorems for weakly interacting infinite systems and the voter model. Ann. Probab. 3 643–663.
  • [9] Kallenberg, O. (1977). Stability of critical cluster fields. Math. Nachr. 77 7–43.
  • [10] Klenke, A. and Mytnik, L. (2008). Infinite rate mutually catalytic branching in infinitely many colonies. Construction, characterization and convergence. Preprint. Available at arXiv:0901.0623 [math.PR].
  • [11] Klenke, A. and Mytnik, L. (2010). Infinite rate mutually catalytic branching. Ann. Probab. 38 1690–1716.
  • [12] Klenke, A. and Oeler, M. (2010). A Trotter-type approach to infinite rate mutually catalytic branching. Ann. Probab. 38 479–497.
  • [13] Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer, New York.
  • [14] Notohara, M. and Shiga, T. (1980). Convergence to genetically uniform state in stepping stone models of population genetics. J. Math. Biol. 10 281–294.
  • [15] Oeler, M. (2008). Mutually catalytic branching at infinite rate. Ph.D. thesis, Univ. Mainz.
  • [16] Protter, P. E. (2004). Stochastic Integration and Differential Equations, 2nd ed. Applications of Mathematics (New York) 21. Springer, Berlin.
  • [17] Shiga, T. (1980). An interacting system in population genetics. J. Math. Kyoto Univ. 20 213–242.