The Annals of Probability
- Ann. Probab.
- Volume 39, Number 1 (2011), 252-290.
On the moments and the interface of the symbiotic branching model
In this paper we introduce a critical curve separating the asymptotic behavior of the moments of the symbiotic branching model, introduced by Etheridge and Fleischmann [Stochastic Process. Appl. 114 (2004) 127–160] into two regimes. Using arguments based on two different dualities and a classical result of Spitzer [Trans. Amer. Math. Soc. 87 (1958) 187–197] on the exit-time of a planar Brownian motion from a wedge, we prove that the parameter governing the model provides regimes of bounded and exponentially growing moments separated by subexponential growth. The moments turn out to be closely linked to the limiting distribution as time tends to infinity. The limiting distribution can be derived by a self-duality argument extending a result of Dawson and Perkins [Ann. Probab. 26 (1998) 1088–1138] for the mutually catalytic branching model.
As an application, we show how a bound on the 35th moment improves the result of Etheridge and Fleischmann [Stochastic Process. Appl. 114 (2004) 127–160] on the speed of the propagation of the interface of the symbiotic branching model.
Ann. Probab., Volume 39, Number 1 (2011), 252-290.
First available in Project Euclid: 3 December 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]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Blath, Jochen; Döring, Leif; Etheridge, Alison. On the moments and the interface of the symbiotic branching model. Ann. Probab. 39 (2011), no. 1, 252--290. doi:10.1214/10-AOP543. https://projecteuclid.org/euclid.aop/1291388302