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 Gärtner (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 Hölder continuous function.

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