Gaussian fluctuation and rate of convergence of the Kardar-Parisi-Zhang equation in weak disorder in $d\geq 3$.

We study the solution $h_\varepsilon$ of the Kardar-Parisi-Zhang (KPZ) equation in $\mathbb R^d\times [0,\infty), d \geq \ 3$: $$ \partial_t h_{\varepsilon} = \frac12 \Delta h_{\varepsilon} + \big[\frac12 |\nabla h_\varepsilon |^2 - C_\varepsilon \big]+ \beta \varepsilon^{\frac{d-2}2} \xi_{\varepsilon} \;,\qquad\,\, h_{\varepsilon}(0,x) =0. $$ Here $\beta>0$ is a parameter called the disorder strength, $\xi_\varepsilon$ is a spatially smoothened (at scale $\varepsilon$) Gaussian space-time white noise and $C_\varepsilon$ is a divergent constant as $\varepsilon\to 0$. For sufficiently small $\beta$, it was shown in [CCM19] that $h_\varepsilon(t,x)- \mathfrak h_\varepsilon(t,x)\to 0$ in probability, where $\mathfrak h_\varepsilon(t,x)$ is also a stationary solution of the KPZ equation (i.e., it is constant in law in $\varepsilon, t,x$) with a random initial condition. In the present article we quantify the exact (polynomial) rate of the above convergence in this regime and show that, the space-time indexed random field $$ \{\mathscr H_\varepsilon(t,x)\}_{t>o,x\in\mathbb R^d}=\big(\varepsilon^{1-\frac d2} [h_\varepsilon(t,x) - \mathfrak h_\varepsilon(t,x)]\big)_{x\in \mathbb R^d, t > 0} $$ converges to a centered Gaussian field $\{\mathscr H(t,x)\}$ with $\mathscr H(t,x)=\gamma(\beta)\int_{[0,\infty)}\int_{\mathbb R^d} \rho(\sigma+t,y-x)\xi(\sigma,z)\mathrm d\sigma\mathrm dz$ where $\rho(t,x)$ is the usual heat kernel. This limit is also the (real-valued) solution of the non-noisy heat equation $\partial_t \mathscr H=\frac 12 \Delta \mathscr H$ with a random initial condition $\mathscr H(0,x)$ given by a Gaussian free field on $\mathbb R^d$. We further obtain convergence in law of the spatial averages $\int_{\mathbb R^d} \mathrm d x \varphi(x) \mathscr H_\varepsilon(t, x) \to \int_{\mathbb R^d} \mathrm d x \varphi(x) \mathscr H(t, x)$ for smooth test functions $\varphi$.