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

Cecile Monthus

TL;DR

This work presents a spectral framework for Markov duality using bi-orthogonal left-right eigenbases, showing that dual generators must be isospectral (sharing eigenvalues possibly complex) and that duality functions link corresponding eigenvectors. By reformulating duality as the operator identity $\boldsymbol w^{\dagger} \\boldsymbol \Omega = \\boldsymbol \Omega \\tilde{\boldsymbol w}$ and expanding in eigenbases, the authors derive precise structural constraints on $\\boldsymbol \Omega$ and illustrate with Time-Reversal, Siegmund, and Moment-Dualities. The Moment-Duality between Wright-Fisher diffusion on $[0,1]$ and the Kingman process on $\mathbb N$ is analyzed in detail, with explicit relations between eigenvectors and a generalization to other Pearson diffusions. The paper also shows how spectral data can guide the construction of new dualities, including a directed jump process in eigenspace when eigenvalues are non-degenerate, highlighting a practical path to uncover novel Markov dualities. Overall, the spectral perspective offers a principled route to understanding, extending, and systematizing Markov dualities across diverse configuration spaces and dynamics.

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 $Ω(x,{\tilde x})$ 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.

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

TL;DR

This work presents a spectral framework for Markov duality using bi-orthogonal left-right eigenbases, showing that dual generators must be isospectral (sharing eigenvalues possibly complex) and that duality functions link corresponding eigenvectors. By reformulating duality as the operator identity and expanding in eigenbases, the authors derive precise structural constraints on and illustrate with Time-Reversal, Siegmund, and Moment-Dualities. The Moment-Duality between Wright-Fisher diffusion on and the Kingman process on is analyzed in detail, with explicit relations between eigenvectors and a generalization to other Pearson diffusions. The paper also shows how spectral data can guide the construction of new dualities, including a directed jump process in eigenspace when eigenvalues are non-degenerate, highlighting a practical path to uncover novel Markov dualities. Overall, the spectral perspective offers a principled route to understanding, extending, and systematizing Markov dualities across diverse configuration spaces and dynamics.

Abstract

The notion of Markov duality between two Markov processes that can live in two different configurations spaces 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 that may be complex, while the duality function 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 and the Kingman Markov jump process on the semi-infinite lattice 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 , whose transitions rates are the opposite-eigenvalues.

Paper Structure

This paper contains 38 sections, 129 equations.

Table of Contents

  1. Introduction
  2. Spectral analysis of the Markov duality between two generators
  3. Notations for the generators and for the propagators of the two Markov processes
  4. Markov matrix $w(x,y)=\langle x \vert {\bold w} \vert y \rangle$ for Markov jump processes in a discrete configuration space
  5. Fokker-Planck differential generator for diffusion processes on the interval $x \in [0,L]$
  6. Notations for the remainder of the present section
  7. Markov duality as an operator identity involving the two Markov generators
  8. Reformulation of the duality of Eq. \ref{['defdualityfonction']} as an operator identity involving the two Markov generators
  9. Markov duality in terms of the spectral decompositions of the two generators ${\bold w}$ and ${\tilde{\bold w}}$
  10. Spectral decomposition of the Markov generator ${\bold w}$ in the bi-orthogonal basis of its left and right eigenvectors
  11. Spectral decomposition of the Markov generator ${\tilde{\bold w}}$ in the bi-orthogonal basis of its left and right eigenvectors
  12. Spectral characterization of the possible dual matrices ${\bold \Omega}$
  13. Duality functions between isospectral Markov models involving only non-vanishing coefficients $\omega_{E,E^*} \ne 0$
  14. Choice of normalization for the eigenvectors to have all coefficients equal to unity $\omega_{E,E^*} =1$
  15. Spectral interpretation of known Markov dualities in the same configuration space
  16. ...and 23 more sections