Multiple Scale Analysis of Spatial Branching Processes under the Palm Distribution
Anita Winter, Universität Erlangen-Nürnberg
Abstract
We consider two types of measure-valued branching
processes on the lattice Zd. These are on the one hand side a
particle system, called branching random walk,
and on the other hand its continuous mass analogue, a system
of interacting diffusions also called super random walk.
It is known that the long-term behavior differs sharply in low and high
dimensions: if d<=2 one gets local extinction,
while, for d>=3, the systems tend to a non-trivial equilibrium.
Due to Kallenberg's criterion, local
extinction goes along with clumping around a 'typical surviving particle.'
This phenomenon is called clustering.
A detailed description of the clusters has been given for
the corresponding processes on R2 in Klenke (1997).
Klenke proved that with the right scaling the mean number of particles
over certain blocks are asymptotically jointly distributed
like marginals of a system of coupled Feller diffusions,
called system of tree indexed Feller diffusions, provided that the
initial intensity is appropriately increased to counteract
the local extinction.
The present paper takes different remedy against the local
extinction allowing also for state-dependent branching mechanisms.
Instead of increasing the initial intensity,
the systems are described
under the Palm distribution. It will turn out together with the
results in Klenke (1997)
that the change to the Palm measure and the
multiple scale analysis commute, as ttoinfty.
The method
of proof is based on the fact that the tree indexed systems of
the branching processes and of the diffusions
in the limit are completely characterized by
all their moments. We develop a machinery to describe the space-time moments
of the superprocess effectively and explicitly.
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