Abstract
We construct the exponential Bernstein inequality for normed fluctuations of a solution of the Dirichlet problem with rapidly oscillating periodic random coefficients with respect to a solution of the averaged Dirichlet problem.
Similar content being viewed by others
References
M. I. Freidlin, “Dirichlet problem for an equation with periodic coefficients dependent on a small parameter,”Teor. Veroyatn. Prim.,9, No. 1, 133–139 (1964).
S. M. Kozlov, “Averaging of random structures,”Dokl. Akad. Nauk SSSR,241, No. 5, 1016–1019 (1978).
V. V. Yurinskii, “On the error of averaging of multidimensional diffusion,”Teor. Veroyatn, Prim.,33, No. 1, 14–24 (1988).
Yu. V. Zhaurov, “Asymptotics of fluctuations in the averaging scheme for elliptic equations with random coefficients,”Teor. Sluch. Prots., No. 11, 31–38 (1983).
A. V. Pozhidaev, “Convergence rate in the averaging principle for elliptic equations with random coefficients,”Teor. Sluch. Prots., No. 12, 59–63 (1984).
A. V. Pozhidaev and V. V. Yurinskii, “On the error of averaging of symmetric elliptic systems,”Izv. Akad. Nauk SSSR, Ser. Mat.,53, No. 4, 851–867 (1989).
N. S. Bakhvalov and G. P. Panasenko,Averaging of Processes in Periodic Systems [in Russian], Nauka, Moscow 1984.
H. C. Bensoussan, J. L. Lions, and G. Papanicolaou,Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam 1978.
B. V. Bondarev, “Averaging in periodic systems under weakly dependent random influences. I,”Teor. Veroyatn. Mat. Statist., No. 5, 12–20 (1991).
I. A. Ibragimov and Yu. V. Linnik,Independent and Stationary Connected Quantities [in Russian], Nauka, Moscow 1965.
O. A. Safonova, “On the asymptotic behavior of integral functionals of diffusive processes with periodic coefficients,”Ukr. Mat. Zh.,44, No. 2, 245–252 (1992).
D. O. Chikin, “Functional limit theorem for stationary processes. Martingale approach,”Teor. Veroyatn. Prim.,34, No. 4, 731–741 (1989).
J. Sam Lazaro and P. A. de’Meyer, “Questions de théorie des flots,”Lect. Notes Math., B465, 1–96 (1975).
P. Protter, “Semimartingales and measure preserving flows,”Ann. Inst. H. Poincaré, Sect. B,22, No. 2, 124–147 (1986).
P. A. Meyer,Probability and Potentials, Blaisdell, Waltham 1966.
A. N. Shiryaev,Probability [in Russian], Nauka, Moscow 1980.
P. P. Zabreiko, A. I. Koshelev, M. A. Krasnosel’skii, et al.,Integral Equations [in Russian], Nauka, Moscow 1968.
R. Z. Khas’minskii,Stability of Systems of Differential Equations under Random Perturbations of Their Parameters [in Russian], Nauka, Moscow 1969.
L. Bers, F. John, and M. Schechter,Partial Differential Equations, Interscience, New York 1964.
I.I. Gikhman and A. V. Skorokhod,Stochastic Differential Equations and Their Applications [in Russian], Naukova Dumka, Kiev 1982.
Rights and permissions
About this article
Cite this article
Bondarev, B.V., Korolev, M.E. Bernstein inequality under averaging of elliptic systems in periodic random media. Ukr Math J 48, 1852–1867 (1996). https://doi.org/10.1007/BF02375372
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02375372