REPRESENTATION THEOREMS FOR INTERACTING MORAN MODELS, INTERACTING FISHER--WRIGHT DIFFUSIONS AND APPLICATIONS

Andreas Greven, Vlada Limic, and Anita Winter

We consider spatial 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. Besides the type configuration one is interested in the time-space evolution of genealogies, incoded in the so-called historical process. In this paper the historical process is constructed for both models.

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 by means of a look-down process. This leads to a strong form of duality between interacting Moran models, interacting Fisher-Wright diffusions on one hand and coalescing random walks on the other hand, which implies in particular the classical form of moment duality for interacting Fisher-Wright diffusions.

It is shown 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 spatial 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.