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We demonstrate a Markov duality between the dynamic asymmetric simple exclusion process (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 the 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.

Dynamic ASEP, Duality, and Continuous q −1 -Hermite Polynomials