Probability Theory and Related Fields

, Volume 87, Issue 4, pp 417–458

# A phase transition for the coupled branching process

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

## Summary

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) .

## Keywords

Phase Transition Stochastic Process Random Walk Probability Theory Markov Process
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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