Population Dynamics in Random Environment, Random Walks on Symmetric Group, and Phylogeny Reconstruction

Abstract

This thesis concerns applications of some probabilistic tools to phylogeny reconstruction and population genetics. Modelling the evolution of species by continuous-time random walks on the signed permutation groups, we study the asymptotic medians of a set of random permutations sampled from simple random walks at time 0.25cn, for c> 0. Running k independent random walks all starting at identity, we prove that the medians approximate the ancestor (identity permutation) up to time 0.25n, while there exists a constant c>1 after which the medians loose credibility as an estimator. We study the median of a set of random permutations on the symmetric group endowed with different metrics. In particular, for a special metric of dissimilarity, called breakpoint, where the space is not geodesic, we find a large group of medians of random permutations using the concept of partial geodesics (or geodesic patches). Also, we study the Fleming-Viot process in random environment (FVRE) via martingale and duality methods. We develop the duality method to the case of time-dependent and quenched martingale problems. Using a family of dual processes we prove the convergence of the Moran processes in random environments to FVRE in Skorokhod topology. We also study the long-time behaviour of FVRE and prove the existence of equilibrium for the joint annealed-environment process and prove an ergodic theorem for the latter.