The Annals of Probability

Infinite rate mutually catalytic branching

Achim Klenke and Leonid Mytnik

Full-text: Open access


Consider the mutually catalytic branching process with finite branching rate γ. We show that as γ → ∞, this process converges in finite-dimensional distributions (in time) to a certain discontinuous process. We give descriptions of this process in terms of its semigroup in terms of the infinitesimal generator and as the solution of a martingale problem. We also give a strong construction in terms of a planar Brownian motion from which we infer a path property of the process.

This is the first paper in a series or three, wherein we also construct an interacting version of this process and study its long-time behavior.

Article information

Ann. Probab., Volume 38, Number 4 (2010), 1690-1716.

First available in Project Euclid: 8 July 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J65: Brownian motion [See also 58J65] 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]

Mutually catalytic branching martingale problem strong construction stochastic differential equations


Klenke, Achim; Mytnik, Leonid. Infinite rate mutually catalytic branching. Ann. Probab. 38 (2010), no. 4, 1690--1716. doi:10.1214/09-AOP520.

Export citation


  • [1] Burkholder, D. L. (1977). Exit times of Brownian motion, harmonic majorization, and Hardy spaces. Adv. Math. 26 182–205.
  • [2] Cox, J. T., Dawson, D. A. and Greven, A. (2004). Mutually catalytic super branching random walks: Large finite systems and renormalization analysis. Mem. Amer. Math. Soc. 171 viii+97.
  • [3] Cox, J. T., Klenke, A. and Perkins, E. A. (2000). Convergence to equilibrium and linear systems duality. In Stochastic Models (Ottawa, ON, 1998). CMS Conference Proceedings 26 41–66. Amer. Math. Soc., Providence, RI.
  • [4] Dawson, D. A., Fleischmann, K. and Xiong, J. (2005). Strong uniqueness for cyclically symbiotic branching diffusions. Statist. Probab. Lett. 73 251–257.
  • [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] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Mathematical Library 24. North-Holland, Amsterdam.
  • [8] Klenke, A. (2008). Probability Theory. Springer, London.
  • [9] 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].
  • [10] Klenke, A. and Mytnik, L. (2009). Infinite rate mutually catalytic branching in infinitely many colonies. The longtime behaviour. Preprint. Available at arXiv:0910.4120 [math.PR].
  • [11] Klenke, A. and Oeler, M. (2010). A Trotter type approach to infinite rate mutually catalytic branching. Ann. Probab. 38 479–497.
  • [12] Le Gall, J. F. and Meyre, T. (1992). Points cônes du mouvement brownien plan, le cas critique. Probab. Theory Related Fields 93 231–247.
  • [13] Mytnik, L. (1998). Uniqueness for a mutually catalytic branching model. Probab. Theory Related Fields 112 245–253.
  • [14] Oeler, M. (2008). Mutually catalytic branching at infinite rate. Ph.D. thesis, Univ. Mainz.
  • [15] Rogers, L. C. G. and Williams, D. (1994). Diffusions, Markov Processes, and Martingales. Vol. 1, 2nd ed. Wiley, Chichester.
  • [16] Shiga, T. and Shimizu, A. (1980). Infinite-dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ. 20 395–416.