Siberian Mathematical Journal

, Volume 32, Issue 4, pp 622–631 | Cite as

Asymptotic normality of the error of averaging elliptic boundary problems

  • A. V. Pozhidaev
Article
  • 24 Downloads

Keywords

Boundary Problem Asymptotic Normality Elliptic Boundary Elliptic Boundary Problem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature Cited

  1. 1.
    V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, and Ngoan Kha T'en, “Averaging and G-convergence,” Usp. Mat. Nauk,34, No. 5, 65–133 (1979).Google Scholar
  2. 2.
    V. V. Yurinskii, “Averaging nondivergent equations of second order with random coefficients,” Sib. Mat. Zh.,23, No. 2, 176–188 (1982).Google Scholar
  3. 3.
    V. V. Zhikov and M. M. Sirazhudinov, “Averaging nondivergent elliptic and parabolic second-order operators and stabilization of a solution of the Cauchy problem,” Mat. Sb.,116, No. 2, 166–186 (1981).Google Scholar
  4. 4.
    V. V. Yurinskii, “Averaging an elliptic boundary problem with random coefficients,” Sib. Mat. Zh.,21, No. 3, 209–223 (1980).Google Scholar
  5. 5.
    A. V. Pozhidaev, “Limit behavior of solutions of boundary problems with random coefficients,” Trudy Inst. Mat. Sib. Otd. Akad. Nauk SSSR,5, 86–96 (1985).Google Scholar
  6. 6.
    A. V. Pozhidaev, “Rate of convergence in the averaging principle for elliptic equations with random coefficients,” Teor. Sluchainykh Protsessov, Kiev,12, 59–63 (1984).Google Scholar
  7. 7.
    V. V. Yurinskii, “Averaging symmetric diffusion in a random medium,” Sib. Mat. Zh.,27, No. 4, 167–180 (1986).Google Scholar
  8. 8.
    V. V. Yurinskii, “Averaging diffusion in a random medium,” Trudy Inst. Mat. Sib. Otd. Akad. Nauk SSSR,5, 76–85 (1985).Google Scholar
  9. 9.
    V. V. Yurinskii, “On the homogenization error for boundary problems with highly oscillating random coefficients,” in: First World Congress Bernoulli Soc., Tashkent, Vol. 2 (1986), p. 651.Google Scholar
  10. 10.
    A. V. Pozhidaev and V. V. Yurinskii, “Estimate of the error of averaging for symmetric elliptic systems with random coefficients,” Usp. Mat. Nauk,43, No. 4, 168, 169 (1988).Google Scholar
  11. 11.
    A. V. Pozhidaev and V. V. Yurinskii, “Error of averaging symmetric elliptic systems,” Izv. Akad. Nauk SSSR, Ser. Mat.,53, No. 4, 851–867.Google Scholar
  12. 12.
    R. Z. Khas'minskii, “Stochastic processes defined by differential equations with small parameter,” Teor. Veroyatn. Primen.,11, No. 2, 240–259 (1966).Google Scholar
  13. 13.
    A. N. Borodin, “Limit theorem for solutions of differential equations with random right side,” Teor. Veroyatn. Primen.,22, No. 3, 498–512 (1977).Google Scholar
  14. 14.
    G. C. Papanicolaou and W. Kohler, “Asymptotic theory of mixing stochastic ordinary differential equations,” Commun. Pure Appl. Math.,27, No. 5, 641–668 (1974).Google Scholar
  15. 15.
    Ya. S. Brodskii and B. Ya. Lukacher, “Fluctuations in an averaging scheme for differential equations with random right side,” Teor. Sluchainykh Protsessov, Kiev,12, 8–17 (1984).Google Scholar
  16. 16.
    A. V. Pozhidaev, “Asymptotic normality of solutions of boundary problems with random coefficients,” Sib. Mat. Zh.,23, No. 4, 142–153 (1982).Google Scholar
  17. 17.
    A. V. Pozhidaev, “Fluctuations in an averaging scheme for ellipic boundary problems,” Teor. Veroyatn. Primen.,29, No. 1, 187–188 (1984).Google Scholar
  18. 18.
    Yu. V. Zhaurov, “Fluctuations in an averaging scheme and summation of independent random variables,” Teor. Sluchainykh Protsessov, Kiev,12, 21–27 (1984).Google Scholar
  19. 19.
    R. Figari, E. Orlandi, and G. Papanicolaou, “Mean field and Gaussian approximation for partial differential equations with random coefficients,“ SIAM J. Appl. Math.,42, No. 5, 1069–1077 (1982).Google Scholar
  20. 20.
    V. V. Yurinskii, “Wave propagation in a one-dimensional random medium,” Novosibirsk (1982) (Preprint, Akad. Nauk SSSR, Sibirsk. Otd, Inst. Mat.; No. 9).Google Scholar
  21. 21.
    N. Akanbaev, “Estimate of the error term in the averaging theorem for random parabolic equations,” Izv. Akad. Nauk KazSSR, Ser. Fiz.-Mat., No. 5, 11–15 (1985).Google Scholar
  22. 22.
    A. V. Pozidaev and V. V. Yurinskii, “On homogeneization of partial differential equations with random coefficients,” in: USSR-Japan Symp. Probab. Theory Math. Statist.: Abstr. Commun., Vol. 2, Tbilisi (1982), pp. 161–163.Google Scholar
  23. 23.
    A. V. Pozhidaev, “Asymptotic normality of solutions of parabolic equations with random coefficients,” Trudy. Inst. Mat. Sib. Otd. Akad. Nauk SSSR,5, 170–182 (1984).Google Scholar
  24. 24.
    H. Watanabe, “Averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients,” Probab. Theory Rel. Fields,77, No. 5, 459–378 (1988).Google Scholar
  25. 25.
    A. V. Pozhidaev, “Asymptotic normality of the error of averaging parabolic boundary problems,” Trudy. Inst. Mat. Sib. Otdel. Akad. Nauk SSSR,13, 181–193 (1989).Google Scholar
  26. 26.
    A. V. Pozhidaev, “Limit theorems for solutions of differential equations with random coefficients,” in: Fourth International Vilnius Conference on Probability Theory and Mathematical Statistics: Abstracts [in Russian], Vol. 3 (1985), pp. 43–44.Google Scholar
  27. 27.
    A. V. Pozidaev, “Asymptotic normality of homogeneization error,” in: First World Congress Bernoulli Society, Vol. 2, Tashkent (1986), p. 648.Google Scholar
  28. 28.
    A. V. Pozidaev, “Asymptotic normality of solutions of differential equations with random coefficients,” in: Fifth International Vilnius Conference on Probability Theory and Mathematical Statistics: Abstr. of Commun., Vol. 2, Vilnius (1989), p. 98.Google Scholar
  29. 29.
    A. V. Požidaev, “Differential equations with random coefficients,” in: 14th IFIP Conference on System Modeling and Optimization, Vol. 5, Leipzig (1980), p. 64.Google Scholar
  30. 30.
    A. V. Pozidaev, “Limit theorems for random equations,” in: Third Hungarian Colloquium on Limit Theorems in Probability and Statistics: Abstr., Pecs, Hungary (1989), p. 50.Google Scholar
  31. 31.
    O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1973).Google Scholar
  32. 32.
    E. Stein, Singular Integrals and Differential Properties of Functions [Russian translation], Mir, Moscow (1973).Google Scholar
  33. 33.
    I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Stochastic Processes [in Russian], Nauka, Moscow (1977).Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • A. V. Pozhidaev

There are no affiliations available

Personalised recommendations