Orthogonal stochastic duality functions from Lie algebra representations

TL;DR

This paper develops a method to derive stochastic duality functions for certain Markov processes using Lie algebra representation theory, resulting in orthogonal duality functions expressed via hypergeometric functions.

Contribution

It introduces a novel approach linking Lie algebra representations to stochastic duality functions, providing explicit orthogonal dualities for particle and diffusion processes.

Findings

Derived duality functions from Lie algebra kernels.

Established orthogonality relations for duality functions.

Applied method to specific particle and diffusion processes.

Abstract

We obtain stochastic duality functions for specific Markov processes using representation theory of Lie algebras. The duality functions come from the kernel of a unitary intertwiner between $*$-representations, which provides (generalized) orthogonality relations for the duality functions. In particular, we consider representations of the Heisenberg algebra and $\mathfrak{su}(1,1)$. Both cases lead to orthogonal (self-)duality functions in terms of hypergeometric functions for specific interacting particle processes and interacting diffusion processes.