Duality and Intertwining for discrete Markov kernels: a relation and examples

TL;DR

This paper explores the relationships between duality and intertwining in discrete Markov chains, extending existing theories with new dual concepts and analyzing their properties through examples like birth-death chains and the Moran model.

Contribution

It introduces an ultrametric dual extending Siegmund duals and discusses the sharp dual, enriching the theoretical framework of duality and intertwining in Markov processes.

Findings

Revisited monotone properties of Siegmund duals in birth-death chains

Introduced an ultrametric dual extending Siegmund kernel

Discussed the properties of the sharp dual in detail

Abstract

We work out some relations between duality and intertwining in the context of discrete Markov chains, fixing up the background of previous relations first established for birth and death chains and their Siegmund duals. In view of the results, the monotone properties resulting from the Siegmund dual of birth and death chains are revisited in some detail, with emphasis on the non neutral Moran model. We also introduce an ultrametric type dual extending the Siegmund kernel. Finally we discuss the sharp dual, following closely the Diaconis-Fill study.