# Dynamic ASEP, duality and continuous $q^{-1}$-Hermite polynomials

## TL;DR

This paper establishes a duality between dynamic and standard ASEP models, explores initial data configurations, and uncovers connections to continuous $q^{-1}$-Hermite polynomials, contributing new analytical tools for studying ASEP.

## Contribution

It introduces a Markov duality for dynamic ASEP, explores new initial data types, and links ASEP to continuous $q^{-1}$-Hermite polynomials, expanding analytical methods.

## Key findings

- Duality between dynamic and standard ASEP established
- New initial data types introduced and analyzed
- Connections to continuous $q^{-1}$-Hermite polynomials uncovered

## Abstract

We demonstrate a Markov duality between the dynamic ASEP and the standard ASEP. We then apply this to step initial data, as well as a half-stationary initial data (which we introduce). While investigating the duality for half-stationary initial data, we uncover and utilize connections to continuous $q^{-1}$-Hermite polynomials. Finally, we introduce a family of stationary initial data which are related to the indeterminate moment problem associated with these $q^{-1}$-Hermite polynomials.