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 $(x,\tilde{x})$ 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.

21 pages

References in corpus (15)

• Duality and hidden symmetries in interacting particle systems
• Mapping out of equilibrium into equilibrium in one-dimensional transport models
• Multiple phases in stochastic dynamics: geometry and probabilities
• Theory of classical metastability in open quantum systems
• Duality and hidden equilibrium in transport models
• Revisiting the Ruelle thermodynamic formalism for Markov trajectories with application to the glassy phase of random trap models
• Optimal Diffusion Processes
• Relating absorbing and hard wall boundary conditions for a one-dimensional run-and-tumble particle
• Large deviations for the Pearson family of ergodic diffusion processes involving a quadratic diffusion coefficient and a linear force
• Replica symmetry breaking in trajectory space for the trap model
• Revisiting boundary-driven non-equilibrium Markov dynamics in arbitrary potentials via supersymmetric quantum mechanics and explicit large deviations at various levels
• Siegmund duality for physicists: a bridge between spatial and first-passage properties of continuous and discrete time stochastic processes
• Low-temperature dynamics of Long-Ranged Spin-Glasses : full hierarchy of relaxation times via real-space renormalization
• On the Kemeny time for continuous-time reversible and irreversible Markov processes with applications to stochastic resetting and to conditioning towards forever-survival
• Markov generators as non-hermitian supersymmetric quantum Hamiltonians: spectral properties via bi-orthogonal basis and Singular Value Decompositions