Stochastic Duality and Orthogonal Polynomials

TL;DR

This paper establishes stochastic duality relations for various Markov processes using orthogonal polynomials as duality functions, enabling easier analysis of expectations in complex systems.

Contribution

It introduces a novel framework linking orthogonal polynomials to stochastic duality for a broad class of Markov processes.

Findings

Duality functions are expressed via classical orthogonal polynomials.

Duality measure matches the stationary measure of the process.

Applicable to interacting particle systems and diffusions.

Abstract

For a series of Markov processes we prove stochastic duality relations with duality functions given by orthogonal polynomials. This means that expectations with respect to the original process (which evolves the variable of the orthogonal polynomial) can be studied via expectations with respect to the dual process (which evolves the index of the polynomial). The set of processes include interacting particle systems, such as the exclusion process, the inclusion process and independent random walkers, as well as interacting diffusions and redistribution models of Kipnis-Marchioro-Presutti type. Duality functions are given in terms of classical orthogonal polynomials, both of discrete and continuous variable, and the measure in the orthogonality relation coincides with the process stationary measure.