The Annals of Probability

The scaling limit of the interface of the continuous-space symbiotic branching model

Jochen Blath, Matthias Hammer, and Marcel Ortgiese

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Abstract

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.

Article information

Source
Ann. Probab. Volume 44, Number 2 (2016), 807-866.

Dates
Received: December 2013
Revised: November 2014
First available in Project Euclid: 14 March 2016

Permanent link to this document
http://projecteuclid.org/euclid.aop/1457960384

Digital Object Identifier
doi:10.1214/14-AOP989

Mathematical Reviews number (MathSciNet)
MR3474460

Zentralblatt MATH identifier
1347.60119

Subjects
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]

Keywords
Symbiotic branching model mutually catalytic branching stepping stone model rescaled interface moment duality Meyer–Zheng topology

Citation

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. http://projecteuclid.org/euclid.aop/1457960384.


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