The Annals of Probability
- Ann. Probab.
- Volume 44, Number 2 (2016), 807-866.
The scaling limit of the interface of the continuous-space symbiotic branching model
The continuous-space symbiotic branching model describes the evolution of two interacting populations that can reproduce locally only in the simultaneous presence of each other. If started with complementary Heaviside initial conditions, the interface where both populations coexist remains compact. Together with a diffusive scaling property, this suggests the presence of an interesting scaling limit. Indeed, in the present paper, we show weak convergence of the diffusively rescaled populations as measure-valued processes in the Skorokhod, respectively the Meyer–Zheng, topology (for suitable parameter ranges). The limit can be characterized as the unique solution to a martingale problem and satisfies a “separation of types” property. This provides an important step toward an understanding of the scaling limit for the interface. As a corollary, we obtain an estimate on the moments of the width of an approximate interface.
Ann. Probab. Volume 44, Number 2 (2016), 807-866.
Received: December 2013
Revised: November 2014
First available in Project Euclid: 14 March 2016
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.) 60H15: Stochastic partial differential equations [See also 35R60]
Blath, Jochen; Hammer, Matthias; Ortgiese, Marcel. The scaling limit of the interface of the continuous-space symbiotic branching model. Ann. Probab. 44 (2016), no. 2, 807--866. doi:10.1214/14-AOP989. https://projecteuclid.org/euclid.aop/1457960384