# Factorized duality, stationary product measures and generating functions

## TL;DR

This paper characterizes all factorized duality functions for a broad class of interacting particle and diffusion systems, revealing that only two main families of dualities encompass all cases, using generating functions and stationary measures.

## Contribution

It provides a complete classification of factorized duality functions for interacting particle and diffusion systems, extending previous results and unifying duality concepts.

## Key findings

- All factorized duality functions are classified into two main families.
- The method unifies duality functions for particle and diffusion systems.
- The approach reveals that only these two families cover all cases.

## Abstract

We find all factorized duality functions for a class of interacting particle systems. The functions we recover are self-duality functions for interacting particle systems such as zero-range processes, symmetric inclusion and exclusion processes, as well as duality and self-duality functions for their continuous counterparts. The approach is based on, firstly, a general relation between factorized duality functions and stationary product measures and, secondly, an intertwining relation provided by generating functions. For the interacting particle systems, these self-duality and duality functions turn out to be generalizations of those previously obtained in [9] and, more recently, in [8]. Thus, we discover that only these two families of dualities cover all possible cases. Moreover, the same method discloses all self-duality functions for interacting diffusion systems such as the Brownian energy process, where both the process and its dual are in continuous variables.