Self-duality of Markov processes and intertwining functions

Abstract

We present a theorem which elucidates the connection between self-duality of Markov processes and representation theory of Lie algebras. In particular, we identify sufficient conditions such that the intertwining function between two representations of a certain Lie algebra is the self-duality function of a (Markov) operator. In concrete terms, the two representations are associated to two operators in interwining relation. The self-dual operator, which arise from an appropriate symmetric linear combination of them, is the generator of a Markov process. The theorem is applied to a series of examples, including Markov processes with a discrete state space (e.g. interacting particle systems) and Markov processes with continuous state space (e.g. diffusion processes). In the examples we use explicit representations of Lie algebras that are unitary equivalent. As a consequence, in the discrete setting self-duality functions are given by orthogonal polynomials whereas in the continuous context they are Bessel functions.

18 pages no figures

References in corpus (9)

• Duality and hidden symmetries in interacting particle systems
• Stochastic higher spin vertex models on the line
• Asymmetric stochastic transport models with ${\mathcal{U}}_q(\mathfrak{su}(1,1))$ symmetry
• Self-Duality for the Two-Component Asymmetric Simple Exclusion Process
• Stochastic duality of ASEP with two particle types via symmetry of quantum groups of rank two
• Quantum algebra symmetry of the ASEP with second-class particles
• ASEP(q,j) converges to the KPZ equation
• Superdiffusivity of Asymmetric Energy Model in Dimension One and Two
• Integrable stochastic dualities and the deformed Knizhnik-Zamolodchikov equation

Cited by in corpus (10)

• Non-compact quantum spin chains as integrable stochastic particle processes
• Orthogonal Dualities of Markov Processes and Unitary Symmetries
• Integrable heat conduction model
• Approximate filtering via discrete dual processes
• Spatially Inhomogeneous Populations with Seed-banks: Duality, Existence and Clustering
• Duality relations between spatial birth-death processes and diffusions in Hilbert space
• Orthogonal polynomial duality of a two-species asymmetric exclusion process
• Stochastic duality and eigenfunctions
• Dual process in the two-parameter Poisson-Dirichlet diffusion
• A Generalized Dynamic Asymmetric Exclusion Process: Orthogonal Dualities and Degenerations