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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