SPDE Limit of Weakly Inhomogeneous ASEP

We study ASEP in a spatially inhomogeneous environment on a torus $ \mathbb{T}^{(N)} = \mathbb{Z}/N\mathbb{Z} $ of $ N $ sites. A given inhomogeneity $ \widetilde{a}(x) \in (0,\infty) $, $x \in \mathbb{T} $, perturbs the overall asymmetric jumping rates $ r < \ell \in (0,1) $ at bonds, so that particles jump from site $ x $ to $ x+1 $ with rate $ r\widetilde{a}(x) $ and from $x+1$ to $x$ with rate $ \ell\widetilde{a}(x) $ (subject to the exclusion rule in both cases). Under the limit $ N\to\infty $, we suitably tune the asymmetry $ (\ell-r) $ to zero like $N^{-\frac{1}{2}}$ and the inhomogeneity $ \widetilde{a}(x) $ to unity, so that the two compete on equal footing. At the level of the G\"{a}rtner (or microscopic Hopf--Cole) transform, we show convergence to a new SPDE -- the Stochastic Heat Equation with a mix of spatial and spacetime multiplicative noise. Equivalently, at the level of the height function we show convergence to the Kardar--Parisi--Zhang equation with a mix of spatial and spacetime additive noise. Our method applies to a general class of $ \widetilde{a}(x) $, which, in particular, includes i.i.d., long-range correlated, and periodic inhomogeneities. The key technical component of our analysis consists of a host of estimates on the \emph{kernel} of the semigroup $ \mathcal{Q}(t):=e^{t\mathcal{H}} $ for a Hill-type operator $ \mathcal{H}:= \frac12\partial_{xx} + \mathcal{A}'(x) $, and its discrete analog, where $ \mathcal{A} $ (and its discrete analog) is a generic H\"{o}lder continuous function.