SPDE Limit of Weakly Inhomogeneous ASEP

We study ASEP in a spatially inhomogeneous environment on a torus ${\mathbb{T}}^{(N)}=\mathbb{Z}\mathrm{/}N\mathbb{Z}$ of $N$ sites. A given inhomogeneity $\tilde{a}(x)\in (0,\mathrm{\infty })$, $x\in \mathbb{T}$, perturbs the overall asymmetric jumping rates $r<\mathrm{\ell }\in (0,1)$ at bonds, so that particles jump from site $x$ to $x+1$ with rate $r\tilde{a}(x)$ and from $x+1$ to $x$ with rate $\mathrm{\ell }\tilde{a}(x)$ (subject to the exclusion rule in both cases). Under the limit $N\to \mathrm{\infty }$, we suitably tune the asymmetry $(\mathrm{\ell }-r)$ to zero like $N^{-\frac{1}{2}}$ and the inhomogeneity $\tilde{a}(x)$ 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 $\tilde{a}(x)$, 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) $,anditsdiscreteanalog,where$ $\mathcal{A}$ $ (and its discrete analog) is a generic H"{o}lder continuous function.