The Concept of Duality and Applications to Markov Processes Arising in Neutral Population Genetics Models

Abstract

One possible and widely used definition of the duality of Markov processes employs functions H relating one process to another in a certain way. For given processes X and Y the space U of all such functions H, called the duality space of X and Y, is studied in this paper. The algebraic structure of U is closely related to the eigenvalues and eigenvectors of the transition matrices of X and Y. Often as for example in physics (interacting particle systems) and in biology (population genetics models) dual processes arise naturally by looking forwards and backwards in time. In particular, time-reversible Markov processes are self-dual. In this paper, results on the duality space are presented for classes of haploid and two-sex population models. For example dim U = N+ 3 for the classical haploid Wright-Fisher model with fixed population size N.