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September 2018 A new look at duality for the symbiotic branching model
Matthias Hammer, Marcel Ortgiese, Florian Völlering
Ann. Probab. 46(5): 2800-2862 (September 2018). DOI: 10.1214/17-AOP1240

Abstract

The symbiotic branching model is a spatial population model describing the dynamics of two interacting types that can only branch if both types are present. A classical result for the underlying stochastic partial differential equation identifies moments of the solution via a duality to a system of Brownian motions with dynamically changing colors. In this paper, we revisit this duality and give it a new interpretation. This new approach allows us to extend the duality to the limit as the branching rate $\gamma$ is sent to infinity. This limit is particularly interesting since it captures the large scale behavior of the system. As an application of the duality, we can explicitly identify the $\gamma=\infty$ limit when the driving noises are perfectly negatively correlated. The limit is a system of annihilating Brownian motions with a drift that depends on the initial imbalance between the types.

Citation

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Matthias Hammer. Marcel Ortgiese. Florian Völlering. "A new look at duality for the symbiotic branching model." Ann. Probab. 46 (5) 2800 - 2862, September 2018. https://doi.org/10.1214/17-AOP1240

Information

Received: 1 September 2016; Revised: 1 October 2017; Published: September 2018
First available in Project Euclid: 24 August 2018

zbMATH: 06964349
MathSciNet: MR3846839
Digital Object Identifier: 10.1214/17-AOP1240

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

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 5 • September 2018
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