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.

Highlights

  • 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 can be studied via expectations with respect to the dual process

  • 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

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Summary

Introduction

Duality theory is a powerful tool to deal with stochastic Markov processes by which information on a given process can be extracted from another process, its dual. The link between the two processes is provided by a set of so-called duality functions, i.e. a set of observables that are functions of both processes and whose expectations, with respect to the two randomness, can be placed in a precise relation (see Definition 1.1 below) It is the aim of this paper to enlarge the space of duality functions for a series of Markov processes that enjoy the stochastic duality property. The works [22, 12, 13, 32, 33] further investigate this framework and provide an algebraic approach to Markov processes with duality starting from a Lie algebra in the symmetric case, and its quantum deformation in the asymmetric one In this approach duality functions emerge as the intertwiners between two different representations. We shall show that duality functions can be placed in relation to orthogonal polynomials

Results
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Paper organization
Preliminaries: hypergeometric orthogonal polynomials
The Symmetric Exclusion Process, SEP(j)
Self-duality for SEP(j)
The Symmetric Inclusion Process, SIP(k)
Self-duality for SIP(k)
The Independent Random Walker, IRW
Self-duality of IRW
The Brownian Momentum Proces, BMP
Duality between BMP and SIP( 14 )
The Brownian Energy Process, BEP(k)
Duality between BEP(k) and SIP(k)
The Kipnis-Marchioro-Presutti process, KMP(k)
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