The Annals of Probability

A new look at duality for the symbiotic branching model

Matthias Hammer, Marcel Ortgiese, and Florian Völlering

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

The symbiotic branching model is a spatial population model describing the dynamics of two interacting types that can only branch if both types are present. A classical result for the underlying stochastic partial differential equation identifies moments of the solution via a duality to a system of Brownian motions with dynamically changing colors. In this paper, we revisit this duality and give it a new interpretation. This new approach allows us to extend the duality to the limit as the branching rate $\gamma$ is sent to infinity. This limit is particularly interesting since it captures the large scale behavior of the system. As an application of the duality, we can explicitly identify the $\gamma=\infty$ limit when the driving noises are perfectly negatively correlated. The limit is a system of annihilating Brownian motions with a drift that depends on the initial imbalance between the types.

Article information

Source
Ann. Probab., Volume 46, Number 5 (2018), 2800-2862.

Dates
Received: September 2016
Revised: October 2017
First available in Project Euclid: 24 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1535097640

Digital Object Identifier
doi:10.1214/17-AOP1240

Mathematical Reviews number (MathSciNet)
MR3846839

Zentralblatt MATH identifier
06964349

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.) 60H15: Stochastic partial differential equations [See also 35R60]

Keywords
Symbiotic branching model mutually catalytic branching stepping stone model rescaled interface moment duality annihilating Brownian motions

Citation

Hammer, Matthias; Ortgiese, Marcel; Völlering, Florian. A new look at duality for the symbiotic branching model. Ann. Probab. 46 (2018), no. 5, 2800--2862. doi:10.1214/17-AOP1240. https://projecteuclid.org/euclid.aop/1535097640


Export citation

References

  • [1] Arratia, R. (1981). Limiting point processes for rescalings of coalescing and annihilating random walks on $\textbf{Z}^{d}$. Ann. Probab. 9 909–936.
  • [2] Aurzada, F. and Döring, L. (2011). Intermittency and ageing for the symbiotic branching model. Ann. Inst. Henri Poincaré Probab. Stat. 47 376–394.
  • [3] Blath, J., Döring, L. and Etheridge, A. (2011). On the moments and the interface of the symbiotic branching model. Ann. Probab. 39 252–290.
  • [4] Blath, J., Hammer, M. and Ortgiese, M. (2016). The scaling limit of the interface of the continuous-space symbiotic branching model. Ann. Probab. 44 807–866.
  • [5] Bramson, M. and Griffeath, D. (1980). Clustering and dispersion rates for some interacting particle systems on $\mathbb{Z}^{1}$. Ann. Probab. 8 183–213.
  • [6] Dawson, D. A. and Perkins, E. A. (1998). Long-time behavior and coexistence in a mutually catalytic branching model. Ann. Probab. 26 1088–1138.
  • [7] Donnelly, P., Evans, S. N., Fleischmann, K., Kurtz, T. G. and Zhou, X. (2000). Continuum-sites stepping-stone models, coalescing exchangeable partitions and random trees. Ann. Probab. 28 1063–1110.
  • [8] 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.
  • [9] Döring, L. and Mytnik, L. (2013). Longtime behavior for mutually catalytic branching with negative correlations. In Advances in Superprocesses and Nonlinear PDEs. Springer Proc. Math. Stat. 38 93–111. Springer, New York.
  • [10] Etheridge, A. M. and Fleischmann, K. (2004). Compact interface property for symbiotic branching. Stochastic Process. Appl. 114 127–160.
  • [11] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. Wiley, New York.
  • [12] Evans, S. N. (1997). Coalescing Markov labelled partitions and a continuous sites genetics model withinfinitely many types. Ann. Inst. Henri Poincaré Probab. Stat. 33 339–358.
  • [13] Hammer, M. and Ortgiese, M. (2016). The infinite rate symbiotic branching model: From discrete to continuous space. Preprint. Available at arXiv:1508.07826.
  • [14] Hammer, M., Ortgiese, M. and Völlering, F. (2017). Entrance laws for annihilating Brownian motions. In preparation.
  • [15] Horn, R. A. and Johnson, C. R. (1991). Topics in Matrix Analysis. Cambridge Univ. Press, Cambridge.
  • [16] Karatzas, I. and Shreve, S. E. (1998). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, Berlin.
  • [17] Klenke, A. and Mytnik, L. (2010). Infinite rate mutually catalytic branching. Ann. Probab. 38 1690–1716.
  • [18] 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.
  • [19] Klenke, A. and Mytnik, L. (2012). Infinite rate mutually catalytic branching in infinitely many colonies: The longtime behavior. Ann. Probab. 40 103–129.
  • [20] Kurtz, T. G. (1991). Random time changes and convergence in distribution under the Meyer–Zheng conditions. Ann. Probab. 19 1010–1034.
  • [21] Meyer, P. A. and Zheng, W. A. (1984). Tightness criteria for laws of semimartingales. Ann. Inst. Henri Poincaré Probab. Stat. 20 353–372.
  • [22] Shiga, T. (1988). Stepping stone models in population genetics and population dynamics. In Stochastic Processes in Physics and Engineering (Bielefeld, 1986). Math. Appl. 42 345–355. Reidel, Dordrecht.
  • [23] Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften. 233. Springer, New York.
  • [24] Tribe, R. (1995). Large time behavior of interface solutions to the heat equation with Fisher–Wright white noise. Probab. Theory Related Fields 102 289–311.
  • [25] Tribe, R. and Zaboronski, O. (2011). Pfaffian formulae for one dimensional coalescing and annihilating systems. Electron. J. Probab. 16 2080–2103.