Summary
We consider a particular Markov process η u t on ℕS,S=ℤn. The random variable η u t (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 µ t ) ast→∞ still have particle density ϕη(x)dμ. Forp>p *, µ t ) 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).
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Greven, A. A phase transition for the coupled branching process. Probab. Th. Rel. Fields 87, 417–458 (1991). https://doi.org/10.1007/BF01304274
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DOI: https://doi.org/10.1007/BF01304274
Keywords
- Phase Transition
- Stochastic Process
- Random Walk
- Probability Theory
- Markov Process