×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite