Self-duality of Markov processes and intertwining functions

TL;DR

This paper establishes a theoretical link between self-duality in Markov processes and Lie algebra representation theory, providing conditions under which intertwining functions serve as self-duality functions, with applications to discrete and continuous processes.

Contribution

It introduces a theorem connecting self-duality of Markov processes to Lie algebra representations, offering a unified framework for discrete and continuous cases.

Findings

Self-duality functions correspond to orthogonal polynomials in discrete cases.

In continuous cases, self-duality functions are Bessel functions.

The theorem applies to various Markov processes, including particle systems and diffusion processes.

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.