On the moments and the interface of the symbiotic branching model. (English) Zbl 1219.60082

Blath, Jochen; Döring, Leif; Etheridge, Alison

Ann. Probab. 39, No. 1, 252-290 (2011).

Authors’ abstract: “We introduce a critical curve separating the asymptotic behavior of the moments of the symbiotic branching model, introduced by A. M. Etheridge and K. Fleischmann [Stochastic Process. Appl. 114, No. 1, 127–160 (2004; Zbl 1072.60086)] into two regimes. Using arguments based on two different dualities and a classical result of Spitzer [Trans. Am. Math. Soc. 87, 187–197 (1958; Zbl 0089.13601)] 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 D. A. Dawson and E. A. Perkins [Ann. Probab. 26, No. 3, 1088–1138 (1998; Zbl 0938.60042)] for the mutually catalytic branching model. As an application, we show how a bound on the 35th moment improves the result of A. M. Etheridge and K. Fleischmann [Zbl 1072.60086] on the speed of the propagation of the interface of the symbiotic branching model.”

Reviewer: Valentin Topchii (Omsk)

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords:

symbiotic branching model; mutually catalytic branching; stepping stone model; parabolic Anderson model; moment duality; self-duality; propagation of interface; exit distribution

Citations:

Zbl 1072.60086; Zbl 0089.13601; Zbl 0938.60042

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