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January 2011 On the moments and the interface of the symbiotic branching model
Jochen Blath, Leif Döring, Alison Etheridge
Ann. Probab. 39(1): 252-290 (January 2011). DOI: 10.1214/10-AOP543


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.


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Jochen Blath. Leif Döring. Alison Etheridge. "On the moments and the interface of the symbiotic branching model." Ann. Probab. 39 (1) 252 - 290, January 2011.


Published: January 2011
First available in Project Euclid: 3 December 2010

zbMATH: 1219.60082
MathSciNet: MR2778802
Digital Object Identifier: 10.1214/10-AOP543

Primary: 60K35
Secondary: 60J80

Keywords: exit distribution , moment duality , Mutually catalytic branching , Parabolic Anderson model , propagation of interface , self-duality , Stepping stone model , Symbiotic branching model

Rights: Copyright © 2011 Institute of Mathematical Statistics


Vol.39 • No. 1 • January 2011
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