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) .
|Title of host publication||Lecture Notes in Mathematics|
|Subtitle of host publication||Probability theory and mathematical statistics (Tbilisi, 1982),|
|Number of pages||11|
|Publication status||Published - 1983|