Homogenization of diffusion processes with random stationary coefficients

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) .