## A central limit theorem for the KPZ equation.(English)Zbl 1388.60111

Stationary space-time random fields $$\zeta$$ and $$\{\zeta^{(\varepsilon)}\}_{\varepsilon\in(0,1]}$$ on $$\mathbb R^2$$ are considered and assumed to satisfy the following: $$\zeta$$ is centered, Lipschitz continuous, $$\mathbf E(|\zeta(z)|^p+|\nabla\zeta(z)|^p)<\infty$$ for all $$p>0$$, $$\mathbf E\,(\zeta(0)\zeta(z))$$ integrates to 1 and $$\sigma\{\zeta(z):z\in K_1\}$$ and $$\sigma\{\zeta(z):z\in K_2\}$$ are independent for any two compact sets $$K_1$$ and $$K_2$$ such that $$\text{dist}(K_1,K_2)\geq 1$$. $$\zeta^{(\varepsilon)}$$ is almost surely periodic in space with the period $$1/\varepsilon$$, $$\sup\,\{\mathbf E|\zeta^{(\varepsilon)}(0)|^p:\varepsilon\in(0,1]\}<\infty$$ for every $$p\geq 1$$, $$\sigma\{\zeta^{(\varepsilon)}(z):z\in K_1\}$$ and $$\sigma\{\zeta^{(\varepsilon)}(z):z\in K_2\}$$ are independent for any two sets $$K_1$$ and $$K_2$$ that are periodic in space with the period $$1/\varepsilon$$ and such that $$\text{dist}(K_1,K_2)\geq 1$$, and, for every $$\varepsilon>0$$, there is a coupling of $$\zeta$$ and $$\zeta^{(\varepsilon)}$$ such that, for every $$T>0$$ and every $$\delta>0$$, and $$\zeta^{(\varepsilon)}\to\zeta$$ in $$L^2$$ in a uniform sense and with a rate specified in the paper.
Let $$\lambda\in\mathbb R$$, $$\beta\in(0,1)$$ and let $$h_0^{(\varepsilon)}$$ be smooth functions on the unit circle $$S^1$$ that converge in $$\mathcal C^\beta$$ as $$\varepsilon\to 0$$ to a limit $$h_0\in\mathcal C^\beta$$, let $$\tilde h_\varepsilon$$ satisfy $\partial_t\tilde h_\varepsilon=\partial_x^2\tilde h_\varepsilon+\lambda(\partial_x\tilde h_\varepsilon)^2+\tilde\zeta_\varepsilon$ on $$S^1$$ with $$\tilde h_\varepsilon(0,x)=h_0^{(\varepsilon)}(x)$$ and $$\tilde\zeta_\varepsilon(t,x)=\varepsilon^{-3/2}\zeta^{(\varepsilon)}(t/\varepsilon^2,x/\varepsilon)$$. Then there exist horizontal and vertical velocities $$v_h$$ and $$v_v^{(\varepsilon)}$$, respectively, such that, for every $$T>0$$, the family of random functions $$\tilde h_\varepsilon(t,x-v_ht)-v_v^{(\varepsilon)}t$$ converges in law as $$\varepsilon\to 0$$ to the Hopf-Cole solution of the KPZ equation $\partial_th=\partial_x^2h+\lambda(\partial_xh)^2+\zeta,\qquad h(0,x)=h_0(x)$ in the space $$\mathcal C^\eta([0,T]\times S^1)$$, for any $$\eta\in(0,\frac 12\land\beta)$$. Moreover, $$v_v^{(\varepsilon)}=\lambda\varepsilon^{-1}C_0+2\lambda^2\varepsilon^{-1/2}C_1+\lambda^3c$$, $$v_h=4\lambda^2\hat c$$ where $$C_0$$ and $$\hat c$$ depend only on the second moment of the random field $$\zeta$$ whereas $$C_1$$ depends on its third moment and $$c$$ depends on the second and fourth moments. If $$\mathbf E(\zeta(0,0)\zeta(t,x))$$ is even as a function of $$x$$ then $$\hat c=0$$.

### MSC:

 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35K55 Nonlinear parabolic equations

### Keywords:

KPZ equation; central limit theorem
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