Representation Theorems for Interacting Moran Models, Interacting Fisher-Wrighter Diffusions and Applications

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.