Stochastic PDE limit of the dynamic ASEP

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 $\epsilon$ tending to zero we set $q=e^{-\epsilon }$ and look at fluctuations, space and time in the scales ${\epsilon }^{-1}$, ${\epsilon }^{-2}$ and ${\epsilon }^{-4}$. 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.

63 pages, 1 figure