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March 2016 The scaling limit of the interface of the continuous-space symbiotic branching model
Jochen Blath, Matthias Hammer, Marcel Ortgiese
Ann. Probab. 44(2): 807-866 (March 2016). DOI: 10.1214/14-AOP989

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.

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Jochen Blath. Matthias Hammer. Marcel Ortgiese. "The scaling limit of the interface of the continuous-space symbiotic branching model." Ann. Probab. 44 (2) 807 - 866, March 2016. https://doi.org/10.1214/14-AOP989

Information

Received: 1 December 2013; Revised: 1 November 2014; Published: March 2016
First available in Project Euclid: 14 March 2016

zbMATH: 1347.60119
MathSciNet: MR3474460
Digital Object Identifier: 10.1214/14-AOP989

Subjects:
Primary: 60K35
Secondary: 60H15 , 60J80

Keywords: Meyer–Zheng topology , moment duality , Mutually catalytic branching , rescaled interface , Stepping stone model , Symbiotic branching model

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 2 • March 2016
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