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Longtime Behavior for Mutually Catalytic Branching with Negative Correlations

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Book cover Advances in Superprocesses and Nonlinear PDEs

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 38))

Abstract

In several examples, dualities for interacting diffusion and particle systems permit the study of the longtime behavior of solutions. A particularly difficult model in which many techniques collapse is a two-type model with mutually catalytic interaction introduced by Dawson/Perkins for which they proved under some assumptions a dichotomy between extinction and coexistence directly from the defining equations.In the present chapter we show how to prove a precise dichotomy for a related model with negatively correlated noises. The proof uses moment bounds on exit times of correlated Brownian motions from the first quadrant and explicit second moment calculations. Since the uniform integrability bound is independent of the branching rate our proof can be extended to infinite branching rate processes.

Mathematics Subject Classification (2000): Primary 60J80; Secondary 60J85.

LM is partly supported by the Israel Science Foundation and B. and G. Greenberg Research Fund (Ottawa)

LD was supported by the Fondation Science Matématiques de Paris

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Acknowledgements

The authors thank an anonymous referee for a very careful reading of the manuscript.

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Authors and Affiliations

  1. LPMA, Université Paris VI, Tours 16/26, 4 Place Jussieu, 75005, Paris, France

    Leif Döring

  2. Faculty of Industrial Engineering and Management Technion Israel Institute of Technology, Haifa, 32000, Israel

    Leonid Mytnik

Authors
  1. Leif Döring
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  2. Leonid Mytnik
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Correspondence to Leif Döring .

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Editors and Affiliations

  1. , Department of Mathematics, University of Colorado, Campus Box 395, Boulder, 80309, Colorado, USA

    Janos Englander

  2. , Department of Mathematics, University of Colorado, Mathematics 225, Boulder, 80309, USA

    Brian Rider

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Döring, L., Mytnik, L. (2013). Longtime Behavior for Mutually Catalytic Branching with Negative Correlations. In: Englander, J., Rider, B. (eds) Advances in Superprocesses and Nonlinear PDEs. Springer Proceedings in Mathematics & Statistics, vol 38. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-6240-8_6

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