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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].

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