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

Abstract The notion of Markov duality between two Markov processes that can live in two different configurations spaces <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mrow> <mml:mrow> <mml:mover> <mml:mi>x</mml:mi> <mml:mo stretchy="true">~</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> that may be complex, while the duality function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mrow> <mml:mrow> <mml:mover> <mml:mi>x</mml:mi> <mml:mo stretchy="true">~</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:math> and the Kingman Markov jump process on the semi-infinite lattice <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> , whose transitions rates are the opposite-eigenvalues.