A phase transition for the coupled branching process |
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Authors: | Andreas Greven |
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Institution: | (1) Institut für Mathematische Stochastik, Universität Göttingen, Lotzestrasse 13, W-3400 Göttingen, Federal Republic of Germany |
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Abstract: | Summary We consider a particular Markov process
t
u
on
S
,S=
n
. The random variable
t
u
(x) is interpreted as the number of particles atx at timet. The initial distribution of this process is a translation invariant measure withf (x)d < . The evolution is as follows: At rateb (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, + 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 (
t
µ
) ast![rarr](/content/m371v8282416472t/xxlarge8594.gif) still have particle density ![phiv](/content/m371v8282416472t/xxlarge981.gif) (x)d . Forp>p
*,
t
µ
) converges ast![rarr](/content/m371v8282416472t/xxlarge8594.gif) 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 (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
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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Keywords: | |
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