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We study the solution $h_\varepsilon$ of the Kardar-Parisi-Zhang (KPZ)equation for $d \geq 3$: $$ \frac{\partial}{\partial t} h_{\varepsilon} =\frac12 \Delta h_{\varepsilon} + \bigg[\frac12 |\nabla h_\varepsilon |^2 -C_\varepsilon\bigg]+  \beta \varepsilon^{\frac{d-2}2} \xi_{\varepsilon} $$ with$h_\varepsilon(0,x)=0$. Here $\xi_\varepsilon=\xi\star \phi_\varepsilon$ is aspatially smoothened (at scale $\varepsilon$) Gaussian space-time white noiseand $C_\varepsilon$ is a divergent constant as $\varepsilon\to 0$. When thedisorder $\beta$ is sufficiently small and $\varepsilon\to 0$,$h_\varepsilon(t,x)- \mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x)\to 0$ inprobability where $\mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x)$ is the {\emphstationary solution} of the KPZ equation - more precisely, $\mathfrakh^{\mathrm{st}}_{\varepsilon}$solves the above equation with a random initialcondition (that is independent of the driving noise $\xi$) and its law isconstant in $(\varepsilon,t,x)$. In the present article we quantify the rate ofthe above convergence in this regime and show that the fluctuation {\emphabout} the stationary solution $$ (\varepsilon^{1-\frac d2} [h_\varepsilon(t,x)- \mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x)])_{x,t} $$ converges pointwise(with finite dimensional distributions in space and time) to a Gaussian freefield (GFF) evolved by the deterministic heat equation. We also identify thefluctuations {\it of} the stationary solution itself and show that the rescaledaverages $\int_{\mathbb R^d} {\mathrm d} x \varphi(x) \varepsilon^{1-\frac d2}[\mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x)- \mathbb E(\mathfrakh^{\mathrm{st}}_{\varepsilon}(t,x))]$ converge to that of the {\emph stationarysolution} of the stochastic heat equation with additive noise, but with(random) {\emph GFF marginals} (instead of flat initial condition).

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