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Homogenization of diffusion processes with random stationary coefficients. (English) Zbl 0535.60071

Probability theory and mathematical statistics, Proc. 4th USSR -Jap. Symp., Tbilisi/USSR 1982, Lect. Notes Math. 1021, 507-517 (1983).
[For the entire collection see Zbl 0509.00020.]
Let (\(\Omega\),\({\mathcal F},\mu)\) be a probability space and \(T_ x\), \(x\in R^ n\), be an ergodic flow on \(\Omega\), \(\{U_ x\}\) be a group of unitary operators on \(L^ 2(\Omega)\) induced by \(\{T_ x\}\) and \(D_ i\) be the infinitesimal generator of \(\{U_ x\}\) in the \(x_ i\)- direction with domain \({\mathcal D} (D_ i)\), \(i=1,...,n\). It is proved that the formal operators \[ A=\sum^{n}_{i,j=1}D_ ia_{ij}(\omega)D_ j+\sum^{n}_{i=1}b_ i(\omega)D_ i\quad and\quad B=m(\omega)^{- 1}A \] are homogenizable, i.e. \(\epsilon Z^{\omega}\!_{t/\epsilon^ 2}\) converges in law on \(C_{[0,\infty)}(R^ n)\) to the law of the defined Gaussian diffusion, as \(\epsilon\to 0\), for almost all \(\omega\), where \(Z^{\omega}\!_ t\) is a diffusion with the generator \[ A^{\omega}=\sum^{n}_{i,j=1}\partial /\partial x_ i+a_{ij}(T_ x\omega)\partial /\partial x_ j+\sum^{n}_{i=1}b_ i(T_ x\omega)\partial /\partial x_ i,\quad or\quad B^{\omega}=m(T_ x\omega)^{-1}A^{\omega}, \] respectively, if the following assumptions are satisfied:
1) there exist \(c_{ij}\in H^ 1(\Omega)=\cap^{n}_{i=1}{\mathcal D}(D_ i)\) such that \(b_ i=\sum^{n}_{j=1}D_ jc_{ij}\) and \(| c_{ij}| \leq M\) for some constant M;
2) \(\int_{\Omega}\sum^{n}_{i=1}b_ iD_ i\Phi d\mu =0\) for all \(\Phi \in H^ 1(\Omega);\)
3) for all \(\omega\in \Omega\), \(\xi \in R^ n\) and some constant \(\nu>0 \nu^{-1}| \xi |^ 2\leq \sum^{n}_{i,j=1}a_{ij}(\omega)\xi_ i\xi_ j\leq \nu | \xi |^ 2\), \(a_{ij}=a_{ji};\)
4) for some constant \(k>0\), \(k^{-1}\leq m(\omega)\leq k;\)
5) for almost all \(\omega\), \(a_{ij}(T_ x\omega)\), \(c_{ij}(T_ x\omega)\) are of \(C^ 2\)-class in x, and \(b_ i\in L^{\infty}(\Omega)\), \(i,j=1,...,n\).
Reviewer: B.Grigelionis

MSC:

60J60 Diffusion processes
60F17 Functional limit theorems; invariance principles
60F20 Zero-one laws

Citations:

Zbl 0509.00020