Summary: We study the solution \(h_{\varepsilon}\) of the Kardar-Parisi-Zhang (KPZ) equation in \(\mathbb{R}^d \times [0, \infty)\) for \(d\geq 3\):\[\frac{\partial}{\partial t} h_{\varepsilon} = \frac{1}{2} \Delta h_{\varepsilon} + \left[ \frac{1}{2} | \nabla h_{\varepsilon}|^2 - C_{\varepsilon} \right] + \beta \varepsilon^{\frac{d - 2}{2}} \xi_{\varepsilon}, \quad h_{\varepsilon}(0, x) = 0.\]Here \(\beta > 0\) is a parameter called the disorder strength, \(\xi_{\varepsilon} = \xi \star \phi_{\varepsilon}\) is a spatially smoothened (at scale \(\varepsilon\)) Gaussian space-time white noise and \(C_{\varepsilon} \) is a divergent constant as \(\varepsilon \to 0\). When \(\beta\) is sufficiently small and \(\varepsilon \to 0\), \(h_{\varepsilon}(t,x) - \mathfrak{h}_{\varepsilon}^{\operatorname{st}}(t,x) \to 0\) in probability where \( \mathfrak{h}_{\varepsilon}^{\operatorname{st}}(t,x)\) is thestationary solution of the KPZ equation – more precisely, \( \mathfrak{h}_{\varepsilon}^{\operatorname{st}}(t,x)\) solves the above equation with a random initial condition (that is independent of the driving noise \(\xi \)) and its marginal law is constant in \((\varepsilon, t, x)\). In the present article we quantify the rate of the above convergence in this regime and show that the fluctuations \(( \varepsilon^{1 - \frac{d}{2}} [h_{\varepsilon} ( t,x) - \mathfrak{h}_{\varepsilon}^{\operatorname{st}} ( t,x ) ] )_{x \in \mathbb{R}^{d} , t > 0}\)about the stationary solution converge pointwise (with finite dimensional distributions in space and time) to a Gaussian free field convoluted with the deterministic heat equation. We also identify the fluctuationsof the stationary solution itself and show that the rescaled averages \(\int_{\mathbb{R}^d} \operatorname{d} x \varphi(x) \varepsilon^{1 - \frac{d}{2}} [ \mathfrak{h}_{\varepsilon}^{\operatorname{st}}(t,x) - \mathbb{E} \mathfrak{h}_{\varepsilon}^{\operatorname{st}}(t,x)]\) converge to that of the stationary solution of the stochastic heat equation with additive noise, but with (random) Gaussian free field marginals (instead of flat initial condition).

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