Homogenization of diffusion processes with random stationary coefficients

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

The author studies the homogenization of diffusion processes with stationary random coefficients. This work continues those of G. C. Papanicolaou and S. R. S. Varadhan [Varadhan, Random fields, Vol. I, II (Esztergom, 1979), 835–873, North-Holland, Amsterdam, 1981; MR0712714; Papanicolaou and Varadhan, Statistics and probability: essays in honor of C. R. Rao, 547–552, North-Holland, Amsterdam, 1982; MR0659505] and of V. V. Zhikov, S. M. Kozlov, O. A. Oleinik and Ha Ten Ngoan [Uspekhi Mat. Nauk 34 (1979), no. 5(209), 65–133; MR0562800]. The main result of this paper concerns the homogenizability of the operators A=∑ i,j≤n D i a ij (ω)D j +∑ i≤n b i (ω)D i and B=m(ω) −1 A when the coefficients satisfy the following conditions: (1) there exist bounded c ij ∈H 1 (Ω) such that b i =∑ j≤n D j c ij ; (2) ∫ Ω ∑ i≤n b i D i φdμ=0 for all φ∈H 1 (Ω) ; (3) a ji =a ij and there exists a constant ν>0 such that ν −1 |ξ| 2 ≤∑ i,j≤n a ij (ω)ξ i ξ j ≤ν|ξ| 2 for all ω∈Ω and ξ∈R n ; (4) there exists a constant k>0 such that k −1 ≤m(ω)≤k ; (5) for almost all ω , a ij (T x ω) and c ij (T x ω) are of class C 2 in x and b i ∈L ∞ (Ω) .
Original languageEnglish
Title of host publicationLecture Notes in Mathematics
Subtitle of host publicationProbability theory and mathematical statistics (Tbilisi, 1982),
PublisherSpringer Berlin
Pages507-517
Number of pages11
Volume1021
Publication statusPublished - 1983
Externally publishedYes

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Homogenization
Diffusion Process
Random Coefficients
Coefficient
Random Field
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Statistics
Operator
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Cite this

Osada, H. (1983). Homogenization of diffusion processes with random stationary coefficients. In Lecture Notes in Mathematics: Probability theory and mathematical statistics (Tbilisi, 1982), (Vol. 1021, pp. 507-517). Springer Berlin.

Homogenization of diffusion processes with random stationary coefficients. / Osada, Hirofumi.

Lecture Notes in Mathematics: Probability theory and mathematical statistics (Tbilisi, 1982), . Vol. 1021 Springer Berlin, 1983. p. 507-517.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Osada, H 1983, Homogenization of diffusion processes with random stationary coefficients. in Lecture Notes in Mathematics: Probability theory and mathematical statistics (Tbilisi, 1982), . vol. 1021, Springer Berlin, pp. 507-517.
Osada H. Homogenization of diffusion processes with random stationary coefficients. In Lecture Notes in Mathematics: Probability theory and mathematical statistics (Tbilisi, 1982), . Vol. 1021. Springer Berlin. 1983. p. 507-517
Osada, Hirofumi. / Homogenization of diffusion processes with random stationary coefficients. Lecture Notes in Mathematics: Probability theory and mathematical statistics (Tbilisi, 1982), . Vol. 1021 Springer Berlin, 1983. pp. 507-517
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AB - The author studies the homogenization of diffusion processes with stationary random coefficients. This work continues those of G. C. Papanicolaou and S. R. S. Varadhan [Varadhan, Random fields, Vol. I, II (Esztergom, 1979), 835–873, North-Holland, Amsterdam, 1981; MR0712714; Papanicolaou and Varadhan, Statistics and probability: essays in honor of C. R. Rao, 547–552, North-Holland, Amsterdam, 1982; MR0659505] and of V. V. Zhikov, S. M. Kozlov, O. A. Oleinik and Ha Ten Ngoan [Uspekhi Mat. Nauk 34 (1979), no. 5(209), 65–133; MR0562800]. The main result of this paper concerns the homogenizability of the operators A=∑ i,j≤n D i a ij (ω)D j +∑ i≤n b i (ω)D i and B=m(ω) −1 A when the coefficients satisfy the following conditions: (1) there exist bounded c ij ∈H 1 (Ω) such that b i =∑ j≤n D j c ij ; (2) ∫ Ω ∑ i≤n b i D i φdμ=0 for all φ∈H 1 (Ω) ; (3) a ji =a ij and there exists a constant ν>0 such that ν −1 |ξ| 2 ≤∑ i,j≤n a ij (ω)ξ i ξ j ≤ν|ξ| 2 for all ω∈Ω and ξ∈R n ; (4) there exists a constant k>0 such that k −1 ≤m(ω)≤k ; (5) for almost all ω , a ij (T x ω) and c ij (T x ω) are of class C 2 in x and b i ∈L ∞ (Ω) .

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