  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.
