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Long-term behaviour of a cyclic catalytic branching system

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  • Kliem, S.

Abstract

We investigate the long-term behaviour of a system of SDEs for d>=2 types, involving catalytic branching and mutation between types. In particular, we show that the overall sum of masses converges to zero but does not hit zero in finite time a.s. We shall then focus on the relative behaviour of types in the limit. We prove weak convergence to a unique stationary distribution that does not put mass on the set where at least one of the coordinates is zero. Finally, we provide a complete analysis of the case d=2.

Suggested Citation

  • Kliem, S., 2011. "Long-term behaviour of a cyclic catalytic branching system," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 357-377, February.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:2:p:357-377
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    References listed on IDEAS

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    1. Dawson, Donald A. & Fleischmann, Klaus & Xiong, Jie, 2005. "Strong uniqueness for cyclically symbiotic branching diffusions," Statistics & Probability Letters, Elsevier, vol. 73(3), pages 251-257, July.
    2. He, Hui, 2009. "Strong uniqueness for a class of singular SDEs for catalytic branching diffusions," Statistics & Probability Letters, Elsevier, vol. 79(2), pages 182-187, January.
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    1. He, Hui, 2009. "Strong uniqueness for a class of singular SDEs for catalytic branching diffusions," Statistics & Probability Letters, Elsevier, vol. 79(2), pages 182-187, January.

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