Stochastic duality and eigenfunctions

TL;DR

This paper characterizes duality relations in finite state Markov processes through spectral decompositions, linking shared eigenvalues to duality functions, and applies this to construct duality functions in solvable models.

Contribution

It provides a full spectral characterization of duality relations and constructs explicit duality functions for certain solvable Markov processes.

Findings

Shared eigenvalues imply duality functions from eigenfunctions

Spectral decomposition fully characterizes duality in finite Markov processes

Explicit duality functions are constructed for solvable models

Abstract

We start from the observation that, anytime two Markov generators share an eigenvalue, the function constructed from the product of the two eigenfunctions associated to this common eigenvalue is a duality function. We push further this observation and provide a full characterization of duality relations in terms of spectral decompositions of the generators for finite state space Markov processes. Moreover, we study and revisit some well-known instances of duality, such as Siegmund duality, and extract spectral information from it. Next, we use the same formalism to construct all duality functions for some solvable examples, i.e., processes for which the eigenfunctions of the generator are explicitly known.