The Annals of Probability

A new look at duality for the symbiotic branching model

Matthias Hammer, Marcel Ortgiese, and Florian Völlering

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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.

Article information

Ann. Probab., Volume 46, Number 5 (2018), 2800-2862.

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

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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]

Symbiotic branching model mutually catalytic branching stepping stone model rescaled interface moment duality annihilating Brownian motions


Hammer, Matthias; Ortgiese, Marcel; Völlering, Florian. A new look at duality for the symbiotic branching model. Ann. Probab. 46 (2018), no. 5, 2800--2862. doi:10.1214/17-AOP1240.

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