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On homogenization of time-dependent random flows. (English) Zbl 0996.60040

The authors consider the Itô stochastic differential equation \[ d{\mathbf x}(t;\omega,\sigma)={\mathbf V}(t,{\mathbf x}(t);\omega) dt +\sqrt{2} d{\mathbf w}(t),\quad {\mathbf x}(0)={\mathbf 0}, \] where \({\mathbf V}:{\mathbf R}\times{\mathbf R}^d\times\Omega\to{\mathbf R}^d\) is a zero mean, stationary, ergodic random vector field over a certain probability space \((\Omega,{\mathcal V},P)\). Assume that \({\mathbf V}\) is locally Lipschitz in \({\mathbf x}\) and \({\mathbf V}(t,{\mathbf x};\omega)=\nabla_{\mathbf x} \cdot{\mathbf H}(t,{\mathbf x};\omega)\), i.e., \({\mathbf V}\) is the divergence of some stationary anti-symmetric random field \({\mathbf H}(t,{\mathbf x};\omega).\) The authors prove that the laws of the trajectories \(\varepsilon {\mathbf x}(t/\varepsilon^2)\), \(t\geq 0,\) converge in probability with respect to \(\omega\), as \(\varepsilon\downarrow 0\), to the law of a Brownian motion with a nontrivial covariance matrix.

MSC:

60F17 Functional limit theorems; invariance principles
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
60G44 Martingales with continuous parameter
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