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Averaging of symmetric diffusion in random medium. (English. Russian original) Zbl 0614.60051

Sib. Math. J. 27, 603-613 (1986); translation from Sib. Mat. Zh. 27, No. 4(158), 167-180 (1986).
The accuracy of convergence is studied of the solutions \(u_{\epsilon}\) of the elliptic equations depending on a random parameter \(L_{\epsilon}u_{\epsilon}=f\), \(x\in Q\subseteq {\mathbb{R}}^ d\), \(u_{\epsilon}=g\), \(x\in \partial Q\), \(\epsilon >0\), to the solution of the averaging problem \(\bar LU=f\), \(x\in Q\), \(U=g\), \(x\in \partial Q\) where \(L_{\epsilon}u=(1/2)D_ i(a_{ij}(y,\omega)D_ ju)\), \(D_ i=\partial /\partial x_ i\), \(y=x/\epsilon\), \(\bar Lu=(1/2)\bar a_{ij}D_ iD_ ju\), \(\bar a{}_{ij}\) constants, \(d\geq 3\), and \(L_{\epsilon}\) is approaching \(\bar L\) using the corresponding resolvent operators.
The diffusion coefficients are supposed to be homogeneous, \(a_{ij}(y+z,\omega)=a_{ij}(y,\omega)\), \(z\in {\mathbb{Z}}^ d\), strictly elliptic, \(C_ 1| \xi |^ 2\leq a_{ij}(y,\omega)\xi_ i\xi_ j\leq C_ 2| \xi |^ 2\), \(\xi \in {\mathbb{R}}^ d\), and to satisfy an ergodic property with respect to the random parameter \(\omega\in \Omega\). The main theorem states the following estimate \[ E| u_{\epsilon}-U|^ 2_{\infty}\leq \tilde C\epsilon^{\gamma},\quad \tilde C,\gamma >0 \] where \(| v|_{\infty}=\sup \{| v(x)|\), \(x\in Q\}\).
Reviewer: C.Vârsan

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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[1] S. M. Kozlov, ?Averaging of random operators,? Mat. Sb.,109, No. 2, 188-202 (1979).
[2] V. V. Yurinskii, ?On averaging of elliptic boundary value problem with random coefficients,? Sib. Mat. Zh.,21, No. 3, 209-223 (1980).
[3] V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, and Ha Tien Ngoan, ?Averaging and G-convergence of elliptic operators,? Usp. Mat. Nauk,34, No. 5, 65-133 (1979). · Zbl 0445.35096
[4] V. V. Yurinskii, ?On averaging of diffusion in a random medium,? Teor. Veroyatn. Primen.,29, No. 3, 607 (1984).
[5] V. V. Yurinskii, ?On averaging of diffusion in random medium,? Tr. IM SO AN SSSR,5, 75-85 (1985).
[6] I. P. Kornfel’d, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory [in Russian], Nauka, Moscow (1980).
[7] I. I. Gikhman and A. V. Skorokhod, Introduction to Theory of Random Processes [in Russian], Nauka, Moscow (1977). · Zbl 0429.60002
[8] J. Neveu, Mathematical Foundations of the Calculus of Probability, Holden-Day (1965). · Zbl 0137.11301
[9] D. G. Aronson, ?Nonnegative solutions of linear parabolic equations,? Ann. della Scuola Normale Superiore di Pisa, Ser. III,22, No. 4, 607-694 (1968). · Zbl 0182.13802
[10] Yu. V. Kazachenko and M. I. Yadrenko, ?Local properties of sampling functions of random fields. I,? in: Theory of Probability and Mathematical Statistics [in Russian], No. 14, Vishcha Shkola, Kiev (1976), pp. 53-66.
[11] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press (1968).
[12] I. A. Ibragimov and Yu. V. Linnik, Independent and Stationarily Connected Quantities [in Russian], Nauka, Moscow (1965). · Zbl 0154.42201
[13] S. M. Kozlov, ?Conductivity of two-dimensional random media,? Usp. Mat. Nauk,34, No. 4, 193-194 (1979).
[14] O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. (1968).
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