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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
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