Probability Theory and Related Fields

, Volume 87, Issue 4, pp 417–458 | Cite as

A phase transition for the coupled branching process

Part I: The ergodic theory in the range of finite second moments
  • Andreas Greven


We consider a particular Markov process η t u on ℕ S ,S=ℤ n . The random variable η t u (x) is interpreted as the number of particles atx at timet. The initial distribution of this process is a translation invariant measure μ withfη(x)dμ<∞. The evolution is as follows: At ratebη(x) a particle is born atx but moves instantaneously toy chosen with probabilityq(x, y). All particles at a site die at ratepd withp∈[0, 1],d,∈ ℝ+ and individual particles die independently from each other at rate (1−p)d. Every particle moves independently of everything else according to a continuous time random walk.

We are mainly interested in the caseb=d andn≧3. The process exhibits a phase transition with respect to the parameterp: Forp<p * all weak limit points of ℒ(η t µ ) ast→∞ still have particle density ϕη(x)dμ. Forp>p*, t µ ) converges ast→∞ to the measure concentrated on the configuration identically 0. We calculatep* as well asp (n) , the points with the property that the extremal invariant measures have forp>p (n) infiniten-th moment of η(x) and forp<p (n) finiten-th moment. We show the case 1>p*>p(2)>p(3)≧...≧p (n) >0, p(n)↓0 occurs for suitable values of the other parameters. Forp<p (2) we prove the system has a one parameter set\((v_\rho )_{\rho \varepsilon \mathbb{R}^ + }\) of extremal invariant measures and we determine their domain of attraction. Part I contains statements of all results but only the proofs of the results about the process for values ofp withp<p (2) and the behaviour of then-th moments andp (n) .


Phase Transition Stochastic Process Random Walk Probability Theory Markov Process 
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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Andreas Greven
    • 1
  1. 1.Institut für Mathematische StochastikUniversität GöttingenGöttingenFederal Republic of Germany

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