Annihilating branching processes

Maury Bramson; Ding Wan-ding; Rick Durrett

doi:10.1016/0304-4149(91)90056-I

Annihilating branching processes

Maury Bramson, Ding Wan-ding, Rick Durrett

School of Mathematics

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Abstract

We consider Markov processes η_t ⊂ Z^d in which (i) particles die at rate δ ≥ 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 δ = 0 product measure with density 1 2 is a stationary distribution; we show it is the limit whenever P(η₀≠ ø) = 1. We also show that if δ is small there is a nontrivial stationary distribution, and that for any δ there are most two extremal translation invariant stationary distributions.

Original language	English (US)
Pages (from-to)	1-17
Number of pages	17
Journal	Stochastic Processes and their Applications
Volume	37
Issue number	1
DOIs	https://doi.org/10.1016/0304-4149(91)90056-I
State	Published - Feb 1991

Bibliographical note

Funding Information:
* Partially supported by NSF Grant DMS86-03437. ** This work was done while this author was visiting Cornell and supported government. *** Partially supported by the National Science Foundation, the Army Research Mathematical Sciences lnstitute at Cornell University, and a Guggenheim fellowship.

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abstract = "We consider Markov processes ηt ⊂ Zd in which (i) particles die at rate δ ≥ 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 δ = 0 product measure with density 1 2 is a stationary distribution; we show it is the limit whenever P(η0≠ {\o}) = 1. We also show that if δ is small there is a nontrivial stationary distribution, and that for any δ there are most two extremal translation invariant stationary distributions.",

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AU - Wan-ding, Ding

AU - Durrett, Rick

N1 - Funding Information: * Partially supported by NSF Grant DMS86-03437. ** This work was done while this author was visiting Cornell and supported government. *** Partially supported by the National Science Foundation, the Army Research Mathematical Sciences lnstitute at Cornell University, and a Guggenheim fellowship.

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N2 - We consider Markov processes ηt ⊂ Zd in which (i) particles die at rate δ ≥ 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 δ = 0 product measure with density 1 2 is a stationary distribution; we show it is the limit whenever P(η0≠ ø) = 1. We also show that if δ is small there is a nontrivial stationary distribution, and that for any δ there are most two extremal translation invariant stationary distributions.

AB - We consider Markov processes ηt ⊂ Zd in which (i) particles die at rate δ ≥ 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 δ = 0 product measure with density 1 2 is a stationary distribution; we show it is the limit whenever P(η0≠ ø) = 1. We also show that if δ is small there is a nontrivial stationary distribution, and that for any δ there are most two extremal translation invariant stationary distributions.

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Annihilating branching processes

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