ON THE MOMENTS AND THE INTERFACE OF THE SYMBIOTIC BRANCHING MODEL

• 成果类型: Article
• 署名作者: Blath, Jochen; Doering, Leif; Etheridge, Alison
• 署名单位: Technical University of Berlin; University of Oxford
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP543
• 发表日期: 2011
• 页码: 252-290
• 关键词: long-time behavior; uniqueness

摘要

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.