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Annihilating branching processes

Author

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  • Bramson, Maury
  • Wan-ding, Ding
  • Durrett, Rick

Abstract

We consider Markov processes [eta]t [subset of] d in which (i) particles die at rate [delta] [greater-or-equal, slanted] 0, (ii) births from x to a neighboring y occur at rate 1, and (iii) when a new particle lands on an occupied site the particles annihilate each other and a vacant site results. When [delta] = 0 product measure with density is a stationary distribution; we show it is the limit whenever P([eta]0[not equal to] ø) = 1. We also show that if [delta] is small there is a nontrivial stationary distribution, and that for any [delta] there are most two extremal translation invariant stationary distributions.

Suggested Citation

  • Bramson, Maury & Wan-ding, Ding & Durrett, Rick, 1991. "Annihilating branching processes," Stochastic Processes and their Applications, Elsevier, vol. 37(1), pages 1-17, February.
    • Handle: RePEc:eee:spapps:v:37:y:1991:i:1:p:1-17
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    Cited by:

    1. Sudbury, Aidan, 1997. "The convergence of the biased annihilating branching process and the double-flipping process in d," Stochastic Processes and their Applications, Elsevier, vol. 68(2), pages 255-264, June.
    2. Aidan Sudbury, 2000. "Dual Families of Interacting Particle Systems on Graphs," Journal of Theoretical Probability, Springer, vol. 13(3), pages 695-716, July.
    3. Alili, Smail & Ignatiouk-Robert, Irina, 2001. "On the surviving probability of an annihilating branching process and application to a nonlinear voter model," Stochastic Processes and their Applications, Elsevier, vol. 93(2), pages 301-316, June.
    4. Jan Niklas Latz & Jan M. Swart, 2023. "Commutative Monoid Duality," Journal of Theoretical Probability, Springer, vol. 36(2), pages 1088-1115, June.

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