## 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

### Keywords:

homogenization; ergodic flow; Gaussian diffusion

Zbl 0509.00020