# Stochastic PDE limit of the dynamic ASEP

## TL;DR

This paper demonstrates that under very weak asymmetry scaling, the height function of the dynamic ASEP converges to a space-time Ornstein-Uhlenbeck process, extending results to ring geometries with generalized rates.

## Contribution

It establishes the stochastic PDE limit of the dynamic ASEP height function under weak asymmetry, including on a ring with generalized jump rates.

## Key findings

- Convergence of dynamic ASEP height function to Ornstein-Uhlenbeck process
- Extension of results to ring geometry with periodic boundary conditions
- Analysis under very weak asymmetry scaling for both standard and generalized rates

## Abstract

We study a stochastic PDE limit of the height function of the dynamic asymmetric simple exclusion process (dynamic ASEP). A degeneration of the stochastic Interaction Round-a-Face (IRF) model of arXiv:1701.05239, dynamic ASEP has a jump parameter $q\in (0,1)$ and a dynamical parameter $\alpha>0$. It degenerates to the standard ASEP height function when $\alpha$ goes to $0$ or $\infty$. We consider very weakly asymmetric scaling, i.e., for $\varepsilon$ tending to zero we set $q=e^{-\varepsilon}$ and look at fluctuations, space and time in the scales $\varepsilon^{-1}$, $\varepsilon^{-2}$ and $\varepsilon^{-4}$. We show that under such scaling the height function of the dynamic ASEP converges to the solution of the space-time Ornstein-Uhlenbeck process. We also introduce the dynamic ASEP on a ring with generalized rate functions. Under the very weakly asymmetric scaling, we show that the dynamic ASEP (with generalized jump rates) on a ring also converges to the solution of the space-time Ornstein-Uhlenbeck process on $[0,1]$ with periodic boundary conditions.