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Law of large numbers and central limit theorem for randomly forced PDE’s. (English) Zbl 1099.35188
The author considers the class of dissipative partial differential equations perturbed by an external random force. In particular it is studied an evolution equation in the space \(H\) of divergence-free vector fields \(u\in L^2(D,\mathbb R^2)\) whose normal component vanishes at \(\partial D\): \(\dot u+Lu+B(u,u)=\eta(t)\). Here \(L\) is the Stokes operator and \(B\) is a bilinear form. It is assumed that the right-hand side \(\eta\) is a random process of the form \(\eta(t)=\sum_{j=1}^{\infty}b_{j}\dot\beta_{j}(t)e_{j}\), where \(b_{j}\geq0\) are some constants such that \(\sum_{j}b_{j}^2<\infty\); \(e_{j}\) is a complete set of normalized eigenfunctions of \(L\), and \(\{\beta_{j}\}\) is a sequence of independent standard Brownian motions.
The following statements are a simplified version of the main results of the paper. Suppose that the non-degeneracy condition \(b_{j}\neq0\) for \(j=1,\ldots,N\) is satisfied for a sufficiently large \(N\). Then for any uniformly Lipschitz bounded functional \(f: H\to \mathbb R\) and any solution \(u(t)\) of the considered equation with deterministic initial condition the following assertions hold. For any \(\varepsilon>0\) there is an a.s. finite random constant \(T\geq1\) such that \[ \left| {1\over t}\int_{0}^{t}f(u(s))\,ds-(f,u)\right| \leq \text{const}\;t^{-1/2+\varepsilon}\;\text{ for}\;t\geq T. \] If \((f,\mu)=0\), then there is a constant \(\sigma\geq0\) depending only on \(f\) such that, for any \(\varepsilon>0\), we have \[ \sup_{z\in R}\left(\theta_{\sigma}(z)\left| P\left\{{1\over\sqrt{t}}\int_{0}^{t}f(u(s))\,ds\leq z\right\}-\Phi_{\sigma}(z)\right| \right)\leq \;\text{ const}\;t^{-1/4+\varepsilon} \;\text{ for}\;t\geq1, \] where \(\theta_{\sigma}\equiv1\) for \(\sigma>0\), \(\theta_0(z)=1\wedge| z| \), and \(\Phi_{\sigma}(z)\) is the centered Gaussian distribution function with variance \(\sigma\).

35R60 PDEs with randomness, stochastic partial differential equations
60F05 Central limit and other weak theorems
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35Q30 Navier-Stokes equations
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