Generalized stepping stone model with $\Xi$-resampling mechanism

TL;DR

This paper introduces a generalized two-dimensional stepping stone model incorporating a $\\Xi$-resampling mechanism, extending classical models by replacing Kingman's coalescent with a more general coalescent process, and analyzes its properties.

Contribution

It constructs and characterizes a new stochastic process with a $\Xi$-resampling mechanism, extending classical stepping stone models with a dual process and stationary distribution analysis.

Findings

Existence of the generalized model proved via moment duality.

Stationary distribution characterized and shown to be non-reversible under certain conditions.

The model generalizes classical stepping stone models with a $\Xi$-coalescent mechanism.

Abstract

A generalized stepping stone model with $\Xi$-resampling mechanism is a two dimensional probability-measure-valued stochastic process whose moment dual is similar to that of the classical stepping stone model except that Kingman's coalescent is replaced by $\Xi$-coalescent. We prove the existence of such a process by specifying its moments using the dual function-valued $\Xi$-coalescent process with geographical labels and migration, and then verifying a multidimensional Hausdorff moment problem. We also characterize the stationary distribution of the generalized stepping stone model and show that it is not reversible if the mutation operator is of uniform jump-type.