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Homogenization of a diffusion process in a divergence-free random field. (English) Zbl 0653.60047

Let X(t) be a solution of the s.d.e. \[ X(t)=\int^{t}_{0}\theta (X(s))ds+\sqrt{2}\beta (t). \] \(\beta\) (\(\cdot)\) is a standard Brownian motion in R d and \[ \theta (x)=\{\theta_ k(x),\quad k=1,...,d\},\quad \theta_ k(x)=\sum^{d}_{j=1}\nabla_ jH_{jk}(x), \] where \(H_{kj}(x)\) is some stationary, sufficiently smooth zero-mean random field, \(H_{kj}(x)=-H_{jk}(x)\), \(x\in R\) d. If H(\(\cdot)\) and \(\theta\) (\(\cdot)\) are square integrable and satisfy certain pathwise regularity and growth conditions, then the processes \(X_{\delta}(\cdot)\) converge in the limit as \(\delta\) \(\to 0\) to some Gaussian diffusion with drift 0 and diffusion matrix \(A=\mathbf{1} +D\). If in addition H is L p-integrable for any \(p<\infty\), then the effective diffusion matrix \(A^{\alpha}\) corresponding to the field \(\alpha\) \(\cdot H(\cdot)\) has an asymptotic expansion of the form \[ A^{\alpha}=\mathbf{1} +D^{\alpha}=\text\textbf{1} +\sum^{m}_{r=1}\alpha^{2r}D^{(2r)}+O(\alpha^{2m+2}),\quad as\quad \alpha \to 0. \] If H(\(\cdot)\) is Gaussian, then the matrices \(D^{(2r)}\) can be evaluated in terms of the spectral density of \(\theta\) (\(\cdot)\).
Reviewer: A.D.Borisenko

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60F05 Central limit and other weak theorems
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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