Haldane linearisation done right: Solving the nonlinear recombination equation the easy way

The nonlinear recombination equation from population genetics has a long history and is notoriously difficult to solve, both in continuous and in discrete time. This is particularly so if one aims at full generality, thus also including degenerate parameter cases. Due to recent progress for the continuous time case via the identification of an underlying stochastic fragmentation process, it became clear that a direct general solution at the level of the corresponding ODE itself should also be possible. This paper shows how to do it, and how to extend the approach to the discrete-time case as well.Comment: 12 pages, 1 figure; some minor update

Recombination models forward and backward in time

Esser M. Recombination models forward and backward in time. Bielefeld: Universität Bielefeld; 2017

A probabilistic analysis of a discrete-time evolution in recombination

We study the discrete-time evolution of a recombination transformation in population genetics. The transformation acts on a product probability space, and its evolution can be described by a Markov chain on a set of partitions that converges to the finest partition. We describe the geometric decay rate to this limit and the quasi-stationary behavior of the Markov chain when conditioned on the event that the chain does not hit the limit.CMM Basal CONICYT Project PB-0

Corrigendum to “A probabilistic analysis of a discrete-time evolution in recombination” (A probabilistic analysis of a discrete-time evolution in recombination (2017) 91 (115–136), (S0196885817300933), (10.1016/j.aam.2017.06.004))

In the paper ‘A probabilistic analysis of a discrete-time evolution in recombination’ [4] the evolution of the recombination transformation Ξ=∑δρδ⨂J∈δμJ was described by a Markov chain (Yn) on a set of partitions, which converges to the finest partition. Our main results were the description of the geometric decay rate to the limit and the quasi-stationary behavior of the Markov chain when conditioned on the event that the chain does not hit the limit. All these results continue to be true, but the Markov chain (Yn) that was claimed to satisfy Ξn=E(⨂J∈Yn μJ) required to be modified. This is done in this Corrigendum