A new look at duality for the symbiotic branching model

Matthias Hammer, Marcel Ortgiese, Florian Vollering

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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 γ 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 γ = ∞ 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.

Original languageEnglish
Pages (from-to)2800-2862
Number of pages63
JournalAnnals of Probability
Volume46
Issue number5
Early online date24 Aug 2018
DOIs
Publication statusPublished - 30 Sep 2018

Keywords

  • Annihilating Brownian motions
  • Moment duality
  • Mutually catalytic branching
  • Rescaled interface
  • Stepping stone model
  • Symbiotic branching model

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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