Representation Theorems for Interacting Moran Models, Interacting Fisher-Wrighter Diffusions and Applications
Andreas Greven, University of Erlangen-Nuernberg
Vlada Limic, University of British Columbia
Anita Winter, University of Erlangen-Nuernberg
Abstract
We consider spatially interacting Moran models and their diffusion limit which are
interacting Fisher-Wright diffusions.
The Moran model is a spatial population model with individuals of different
type located on sites given by elements of an Abelian group.
The dynamics of the system consists of independent migration of individuals
between the sites and a resampling mechanism at each site, i.e.,
pairs of individuals are replaced by new pairs where
each newcomer takes the type of a randomly
chosen individual from the parent pair.
Interacting Fisher-Wright diffusions collect the relative frequency
of a subset of types evaluated for the separate sites in
the limit of infinitely many individuals per site.
One is interested in the type
configuration as well as the time-space evolution of
genealogies, encoded in the so-called historical process.
The first goal of the paper is the analytical characterization of
the historical processes for both models as solutions of well-posed
martingale problems and the development of a corresponding duality theory.
For that purpose, we link both the historical Fisher-Wright diffusions
and the historical Moran models by the so-called look-down
process. That is,
for any fixed time, a collection of historical Moran models
with increasing particle intensity and a particle
representation for the limiting historical
interacting Fisher-Wright diffusions are provided
on one and the same probability space.
This leads to a strong form of duality
between spatially interacting Moran models, interacting Fisher-Wright
diffusions on the one hand and coalescing random walks on the other
hand, which extends
the classical weak form of moment duality
for interacting Fisher-Wright diffusions.
Our second goal is to show
that this representation can be used to obtain new results on
the long-time behavior,
in particular (i) on the structure of the equilibria,
and of the equilibrium historical processes, and (ii)
on the behavior of our
models on large but finite site space in comparison
with our models on infinite site space.
Here the so-called finite system scheme is established for spatially interacting Moran
models which implies via the look-down representation
also the already known results for interacting
Fisher-Wright diffusions. Furthermore suitable versions of the finite system
scheme on the level of historical processes are newly developed and verified.
In the long run the provided
look-down representation is intended to answer
questions about finer path properties of interacting
Fisher-Wright diffusions.
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