Mutually catalytic branching in the plane: uniqueness

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Abstract

We study a pair of populations in R2 which undergo diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. Previous work had established the existence of such a process and derived some of its small scale and large scale properties. This paper is primarily focused on the proof of uniqueness of solutions to the martingale problem associated with the model. The self-duality property of solutions, which is crucial for proving uniqueness and was used in the previous work to derive many of the qualitative properties of the process, is also established.

Résumé

On étudie un couple de populations dans le plan, sujettes à des phénomènes de branchement et de diffusion.

Le taux de branchement de chaque type est supposé proportionnel à ka densité de l'autre type. L'existence et les propriétés à grande et petite échelle d'un tel processus a été établie dans un article précédent. Ce travail est centré sur l'unicité de la solution du problème des martingales associé au modèle.

La propriété d'auto-dualité, qui est cruciale dans la démonstration de l'unicité et a été employée dans le travail précédent pour établir de nombreuses propriétés qualitatives du processus, est également démontrée.

References (16)

  • M. Barlow et al.

    Collision local times and measure-valued processes

    Can. J. Math.

    (1991)
  • D. Dawson

    Measure-valued Markov Processes, École d'été de Probabilités de Saint Flour

    (1991)
  • D. Dawson et al.

    Mutually catalytic branching in the plane: Finite measure states

    Ann. Probab.

    (2002)
  • D. Dawson et al.

    Mutually catalytic branching in the plane: infinite measure states

    Electron. J. Probab.

    (2002)
  • D. Dawson et al.

    Long time behaviour and co-existence in a mutually catalytic branching model

    Ann. Probab.

    (1998)
  • P. Donnelly et al.

    Particle representations for measure-valued population models

    Ann. Probab.

    (1999)
  • S.N. Ethier et al.

    Markov Process: Characterization and Convergence

    (1986)
  • S. Evans et al.

    Collision local times, historical stochastic calculus, and competing superprocesses

    Electron. J. Probab.

    (1998)
There are more references available in the full text version of this article.

Cited by (10)

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1

Supported in part by an NSERC grant and a Max Planck Award.

2

Supported in part by the DFG.

3

Supported in part by the US–Israel Binational Science Foundation (grant No. 2000065) and the Israel Science Foundation (grant No. 116/01-10.0).

4

Supported in part by an NSERC grant.

5

Supported in part by UT-ORNL Science Alliance.

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