Space-time fluctuation of the Kardar-Parisi-Zhang equation in $d\ge 3$ and the Gaussian free field

Abstract

We study the solution $h_{\epsilon }$ of the Kardar-Parisi-Zhang (KPZ) equation for $d\ge 3$: $$\frac{\partial }{\partial t}{h}_{\epsilon }=\frac{1}{2}\Delta h_{\epsilon }+[\frac{1}{2}|\nabla h_{\epsilon }{|}^2-C_{\epsilon }]+\beta {\epsilon }^{\frac{d-2}{2}}{\xi }_{\epsilon }$$ with $h_{\epsilon }(0,x)=0$. Here ${\xi }_{\epsilon }=\xi \star {\varphi }_{\epsilon }$ is a spatially smoothened (at scale $\epsilon$) Gaussian space-time white noise and $C_{\epsilon }$ is a divergent constant as $\epsilon \to 0$. When the disorder $\beta$ is sufficiently small and $\epsilon \to 0$, $h_{\epsilon }(t,x)-{\mathfrak{h}}_{\epsilon }^{\mathrm{st}}(t,x)\to 0$ in probability where ${\mathfrak{h}}_{\epsilon }^{\mathrm{st}}(t,x)$ is the {\emph stationary solution} of the KPZ equation - more precisely, ${\mathfrak{h}}_{\epsilon }^{\mathrm{st}}$solves the above equation with a random initial condition (that is independent of the driving noise $\xi$) and its law is constant in $(\epsilon ,t,x)$. In the present article we quantify the rate of the above convergence in this regime and show that the fluctuation {\emph about} the stationary solution $$({\epsilon }^{1-\frac{d}{2}}[h_{\epsilon }(t,x)-{\mathfrak{h}}_{\epsilon }^{\mathrm{st}}(t,x)]{)}_{x,t}$$ converges pointwise (with finite dimensional distributions in space and time) to a Gaussian free field (GFF) evolved by the deterministic heat equation. We also identify the fluctuations {\it of} the stationary solution itself and show that the rescaled averages ${\int }_{{\mathbb{R}}^d}\mathrm{d}x\phi (x){\epsilon }^{1-\frac{d}{2}}[{\mathfrak{h}}_{\epsilon }^{\mathrm{st}}(t,x)-\mathbb{E}({\mathfrak{h}}_{\epsilon }^{\mathrm{st}}(t,x))]$ converge to that of the {\emph stationary solution} of the stochastic heat equation with additive noise, but with (random) {\emph GFF marginals} (instead of flat initial condition).

In the current version, Theorem 2.2 is new (in the earlier version it appeared as Remark 2.8) and gives the fluctuations of the stationary solution. Introduction also revised