Markov dualities via the spectral decompositions of the two Markov generators in their bi-orthogonal basis of right and left eigenvectors

TL;DR

This paper revisits Markov duality through spectral decompositions of generators, revealing new insights into classical dualities and enabling the construction of novel dual processes using eigenvector analysis.

Contribution

It introduces a spectral framework for understanding Markov dualities, connecting eigenvectors to duality functions, and demonstrates how to construct new dual processes from eigenvalues.

Findings

Spectral perspective clarifies classical dualities like time-reversal and Siegmund duality.

Eigenvector analysis links duality functions to right and left eigenvectors of generators.

Constructs new dual processes based on eigenvalues, exemplified with Wright-Fisher and Kingman models.

Abstract

The notion of Markov duality between two Markov processes that can live in two different configurations spaces $(x,{\tilde x})$ is revisited via the spectral decompositions of the two Markov generators in their bi-orthogonal basis of right and left eigenvectors. In this formulation, the two generators should have the same eigenvalues $(-E)$ that may be complex, while the duality function $\Omega(x,{\tilde x})$ can be considered as a mapping between the right and the left eigenvectors of the two models. We describe how this spectral perspective is useful to better understand two well-known dualities between processes defined in the same configuration space: the Time-Reversal duality corresponds to an exchange between the right and the left eigenvectors that involves the steady state, while in the Siegmund duality, the left eigenvectors correspond to integrals of the dual right eigenvectors. We then focus on the famous Moment-Duality between the Wright-Fisher diffusion on the interval $x \in [0,1] $ and the Kingman Markov jump process on the semi-infinite lattice $n \in {\mathbb N}$ in order to analyze the relations between their eigenvectors living in two different configuration spaces. Finally, we discuss how the spectral perspective can be used to construct new dualities and we give an example for the case of non-degenerate real eigenvalues, where one can always construct a dual Directed Jump process on the semi-infinite lattice $n \in {\mathbb N}$, whose transitions rates are the opposite-eigenvalues.