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Homogenization of random parabolic operators. (English) Zbl 0892.35019
Cioranescu, Doina (ed.) et al., Homogenization and applications to material sciences. Proceedings of the international conference, Nice, France, June 6–10, 1995. Tokyo: Gakkotosho. GAKUTO Int. Ser., Math. Sci. Appl. 9, 241-255 (1995).
Summary: The homogenization problem for a random parabolic operator of the following type $A^\varepsilon= {\partial\over \partial t} -{\partial\over \partial x_i}a_{ij} \left({x \over \varepsilon}, \xi_{{t \over \varepsilon^\alpha}} \right) {\partial \over\partial x_j}$ is studied; here $$\varepsilon$$ is a small parameter, $$\alpha>0$$ and $$\xi_s$$ is a diffusion process in $$\mathbb{R}^d$$ possessing an invariant measure with density $$p(y)$$. The matrix $$a_{ij} (z,y)$$ is supposed to be periodic in $$z$$ and uniformly elliptic. It is shown that under some additional assumptions on $$\xi_s$$ the operators $$A^\varepsilon$$ $$G$$-converge as $$\varepsilon\to 0$$ to the specific parabolic operator $$\overline A$$ with constant coefficients. It should be noted that the averaging procedure depends crucially on whether $$\alpha>2$$, $$\alpha=2$$, or $$\alpha<2$$. In particular, for $$\alpha=2$$ the homogenized matrix $$\{\overline a_{ij}\}$$ can be found in terms of joint distribution of the process $$\xi_s$$ and the process ruled by the operator $${\partial \over \partial z_i} a_{ij} (z,y) {\partial\over \partial z_j}$$.
For the entire collection see [Zbl 0873.00028].

##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35R60 PDEs with randomness, stochastic partial differential equations 35K10 Second-order parabolic equations
##### Keywords:
diffusion process; invariant measure; averaging