The Annals of Applied Probability

Finite system scheme for mutually catalytic branching with infinite branching rate

Leif Döring, Achim Klenke, and Leonid Mytnik

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

For many stochastic diffusion processes with mean field interaction, convergence of the rescaled total mass processes towards a diffusion process is known.

Here, we show convergence of the so-called finite system scheme for interacting jump-type processes known as mutually catalytic branching processes with infinite branching rate. Due to the lack of second moments, the rescaling of time is different from the finite rate mutually catalytic case. The limit of rescaled total mass processes is identified as the finite rate mutually catalytic branching diffusion. The convergence of rescaled processes holds jointly with convergence of coordinate processes, where the latter converge at a different time scale.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 5 (2017), 3113-3152.

Dates
Received: October 2015
Revised: September 2016
First available in Project Euclid: 3 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1509696042

Digital Object Identifier
doi:10.1214/17-AAP1277

Mathematical Reviews number (MathSciNet)
MR3719954

Zentralblatt MATH identifier
1379.60097

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J60: Diffusion processes [See also 58J65] 60J75: Jump processes 60F05: Central limit and other weak theorems 60H20: Stochastic integral equations

Keywords
Finite systems scheme interacting diffusions meanfield limit mutually catalytic branching

Citation

Döring, Leif; Klenke, Achim; Mytnik, Leonid. Finite system scheme for mutually catalytic branching with infinite branching rate. Ann. Appl. Probab. 27 (2017), no. 5, 3113--3152. doi:10.1214/17-AAP1277. https://projecteuclid.org/euclid.aoap/1509696042


Export citation

References

  • [1] Aldous, D. (1978). Stopping times and tightness. Ann. Probab. 6 335–340.
  • [2] Baillon, J.-B., Clément, P., Greven, A. and den Hollander, F. (1993). A variational approach to branching random walk in random environment. Ann. Probab. 21 290–317.
  • [3] Baillon, J.-B., Clément, P., Greven, A. and den Hollander, F. (1995). On the attracting orbit of a non-linear transformation arising from renormalization of hierarchically interacting diffusions. I. The compact case. Canad. J. Math. 47 3–27.
  • [4] 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.
  • [5] Cox, J. T. and Greven, A. (1990). On the long term behavior of some finite particle systems. Probab. Theory Related Fields 85 195–237.
  • [6] Cox, J. T., Greven, A. and Shiga, T. (1995). Finite and infinite systems of interacting diffusions. Probab. Theory Related Fields 103 165–197.
  • [7] Dawson, D. A. and Greven, A. (1993). Hierarchical models of interacting diffusions: Multiple time scale phenomena, phase transition and pattern of cluster-formation. Probab. Theory Related Fields 96 435–473.
  • [8] Dawson, D. A., Greven, A., den Hollander, F., Sun, R. and Swart, J. M. (2008). The renormalization transformation of two-type branching models. Ann. Inst. Henri Poincaré Probab. Stat. 44 1038–1077.
  • [9] Dawson, D. A. and Perkins, E. A. (1998). Long-time behavior and coexistence in a mutually catalytic branching model. Ann. Probab. 26 1088–1138.
  • [10] Dellacherie, C. and Meyer, P.-A. (1983). Probabilités et Potentiel. Chapitres IX à XI, revised ed. Publications de l’Institut de Mathématiques de L’Université de Strasbourg [Publications of the Mathematical Institute of the University of Strasbourg] XVIII. Hermann, Paris.
  • [11] Döring, L. and Mytnik, L. (2012). Mutually catalytic branching processes and voter processes with strength of opinion. ALEA Lat. Am. J. Probab. Math. Stat. 9 1–51.
  • [12] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [13] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • [14] Klenke, A. and Mytnik, L. (2010). Infinite rate mutually catalytic branching. Ann. Probab. 38 1690–1716.
  • [15] Klenke, A. and Mytnik, L. (2012). Infinite rate mutually catalytic branching in infinitely many colonies: Construction, characterization and convergence. Probab. Theory Related Fields 154 533–584.
  • [16] Klenke, A. and Oeler, M. (2010). A Trotter-type approach to infinite rate mutually catalytic branching. Ann. Probab. 38 479–497.
  • [17] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École D’été de Probabilités de Saint-Flour, XIV—1984. Lecture Notes in Math. 1180 265–439. Springer, Berlin.