1 Yosuke Kawamoto, Hirofumi Osada, Dynamical Bulk Scaling Limit of Gaussian Unitary Ensembles and Stochastic Differential Equation Gaps, Journal of Theoretical Probability, https://doi.org/10.1007/s10959-018-0816-2, 1-27, 2018.02, The distributions of N-particle systems of Gaussian unitary ensembles converge to Sine(Formula presented.) point processes under bulk scaling limits. These scalings are parameterized by a macro-position (Formula presented.) in the support of the semicircle distribution. The limits are always Sine(Formula presented.) point processes and independent of the macro-position (Formula presented.) up to the dilations of determinantal kernels. We prove a dynamical counterpart of this fact. We prove that the solution to the N-particle system given by a stochastic differential equation (SDE) converges to the solution of the infinite-dimensional Dyson model. We prove that the limit infinite-dimensional SDE (ISDE), referred to as Dyson’s model, is independent of the macro-position (Formula presented.), whereas the N-particle SDEs depend on (Formula presented.) and are different from the ISDE in the limit whenever (Formula presented.).. 2 Hirofumi Osada, Shota Osada, Discrete Approximations of Determinantal Point Processes on Continuous SpacesTree Representations and Tail Triviality, Journal of Statistical Physics, https://doi.org/10.1007/s10955-017-1928-2, 170, 2, 421-435, 2018.01, We prove tail triviality of determinantal point processes μ on continuous spaces. Tail triviality has been proved for such processes only on discrete spaces, and hence we have generalized the result to continuous spaces. To do this, we construct tree representations, that is, discrete approximations of determinantal point processes enjoying a determinantal structure. There are many interesting examples of determinantal point processes on continuous spaces such as zero points of the hyperbolic Gaussian analytic function with Bergman kernel, and the thermodynamic limit of eigenvalues of Gaussian random matrices for Sine 2, Airy 2, Bessel 2, and Ginibre point processes. Our main theorem proves all these point processes are tail trivial.. 3 Hirofumi Osada, Tomoyuki Shirai, Absolute continuity and singularity of Palm measures of the Ginibre point process, Probability Theory and Related Fields, https://doi.org/10.1007/s00440-015-0644-6, 165, 3-4, 725-770, 2016.08, We prove a dichotomy between absolute continuity and singularity of the Ginibre point process G and its reduced Palm measures { Gx, x∈ Cℓ, ℓ= 0 , 1 , 2 … } , namely, reduced Palm measures Gx and Gy for x∈ Cℓ and y∈ Cn are mutually absolutely continuous if and only if ℓ= n; they are singular each other if and only if ℓ≠ n. Furthermore, we give an explicit expression of the Radon–Nikodym density dGx/ dGy for x, y∈ Cℓ.. 4 Hirofumi Osada, Hideki Tanemura, Strong Markov property of determinantal processes with extended kernels, Stochastic Processes and their Applications, https://doi.org/10.1016/j.spa.2015.08.003, 126, 1, 186-208, 2016.01, Noncolliding Brownian motion (Dyson's Brownian motion model with parameter β=2) and noncolliding Bessel processes are determinantal processes; that is, their space-time correlation functions are represented by determinants. Under a proper scaling limit, such as the bulk, soft-edge and hard-edge scaling limits, these processes converge to determinantal processes describing systems with an infinite number of particles. The main purpose of this paper is to show the strong Markov property of these limit processes, which are determinantal processes with the extended sine kernel, extended Airy kernel and extended Bessel kernel, respectively. We also determine the quasi-regular Dirichlet forms and infinite-dimensional stochastic differential equations associated with the determinantal processes.ランダム行列に関係する1次元空間の無限粒子系の運動を記述する確率力学の構成には、確率解析的な手法と、時空間相関関数の計算に基づく代数的な方法がある。この論文は、代数的な手法によって構成された確率力学の強マルコフ性を証明し、筆者達の他の結果と合わせて、これら二つの手法によって構成された確率力学が同一であることを証明した。その結果、解析的手法によって証明されている粒子の運動のパスとしての性質、また、代数的に手法によって証明されている確率力学の定量的な性質が、これらは、それぞれ別の手法ではとr手も証明されないものだが、これら二つの確率力学の同一性が示されたために、両方に対して成立することが判明した。. 5 Hirofumi Osada, Self-diffusion constants of non-colliding interacting Brownian motions in one spatial dimension, RIMS Kôkyûroku Bessatsu, B59, 253-272, 2016. 6 Hirofumi Osada, Hideki Tanemura, Stochastic differential equations related to random matrix theory, RIMS Kôkyûroku Bessatsu, B59, 203-214, 2016. 7 Ryuichi Honda, Hirofumi Osada, Infinite-dimensional stochastic differential equations related to Bessel random point fields, Stochastic Processes and their Applications, https://doi.org/10.1016/j.spa.2015.05.005, 125, 10, 3801-3822, 2015.07, We solve the infinite-dimensional stochastic differential equations (ISDEs) describing an infinite number of Brownian particles in ℝ+ interacting through the two-dimensional Coulomb potential. The equilibrium states of the associated unlabeled stochastic dynamics are Bessel random point fields. To solve these ISDEs, we calculate the logarithmic derivatives, and prove that the random point fields are quasi-Gibbsian.. 8 Hirofumi Osada, Hideki Tanemura, Cores of Dirichlet forms related to random matrix theory, Proceedings of the Japan Academy Series A: Mathematical Sciences, https://doi.org/10.3792/pjaa.90.145, 90, 10, 145-150, 2014.10, We prove the sets of polynomials on configuration spaces are cores of Dirichlet forms describing interacting Brownian motion in infinite dimensions. Typical examples of these stochastic dynamics are Dyson's Brownian motion and Airy interacting Brownian motion. Both particle systems have logarithmic interaction potentials, and naturally arise from random matrix theory. The results of the present paper will be used in a forth coming paper to prove the identity of the infinite-dimensional stochastic dynamics related to the random matrix theories constructed by apparently different methods: the method of space-time correlation functions and that of stochastic analysis.. 9 Hirofumi Osada, Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials IIAiry random point field, Stochastic Processes and their Applications, https://doi.org/10.1016/j.spa.2012.11.002, 123, 3, 813-838, 2013.01, We give a new sufficient condition of the quasi-Gibbs property. This result is a refinement of one given in a previous paper (Osada (in press) [18]), and will be used in a forthcoming paper to prove the quasi-Gibbs property of Airy random point fields (RPFs) and other RPFs appearing under soft-edge scaling. The quasi-Gibbs property of RPFs is one of the key ingredients to solve the associated infinite-dimensional stochastic differential equation (ISDE). Because of the divergence of the free potentials and the interactions of the finite particle approximation under soft-edge scaling, the result of the previous paper excludes the Airy RPFs, although Airy RPFs are the most significant RPFs appearing in random matrix theory. We will use the result of the present paper to solve the ISDE for which the unlabeled equilibrium state is the Airy β RPF with β=1,2,4.. 10 Hirofumi Osada, Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials, Annals of Probability, https://doi.org/10.1214/11-AOP736, 41, 1, 1-49, 2013.01, We investigate the construction of diffusions consisting of infinitely numerous Brownian particles moving in Rd and interacting via logarithmic functions (two-dimensional Coulomb potentials). These potentials are very strong and act over a long range in nature. The associated equilibrium states are no longer Gibbs measures. We present general results for the construction of such diffusions and, as applications thereof, construct two typical interacting Brownian motions with logarithmic interaction potentials, namely the Dyson model in infinite dimensions and Ginibre interacting Brownian motions. The former is a particle system in R, while the latter is in R2. Both models are translation and rotation invariant in space, and as such, are prototypes of dimensions d = 1, 2, respectively. The equilibrium states of the former diffusion model are determinantal or Pfaffian random point fields with sine kernels. They appear in the thermodynamical limits of the spectrum of the ensembles of Gaussian random matrices such as GOE, GUE and GSE. The equilibrium states of the latter diffusion model are the thermodynamical limits of the spectrum of the ensemble of complex non-Hermitian Gaussian random matrices known as the Ginibre ensemble.. 11 Hirofumi Osada, Infinite-dimensional stochastic differential equations related to random matrices, Probability Theory and Related Fields, https://doi.org/10.1007/s00440-011-0352-9, 153, 3-4, 471-509, 2012.08, We solve infinite-dimensional stochastic differential equations (ISDEs) describing an infinite number of Brownian particles interacting via two-dimensional Coulomb potentials. The equilibrium states of the associated unlabeled stochastic dynamics are the Ginibre random point field and Dyson's measures, which appear in random matrix theory. To solve the ISDEs we establish an integration by parts formula for these measures. Because the long-range effect of two-dimensional Coulomb potentials is quite strong, the properties of Brownian particles interacting with two-dimensional Coulomb potentials are remarkably different from those of Brownian particles interacting with Ruelle's class interaction potentials. As an example, we prove that the interacting Brownian particles associated with the Ginibre random point field satisfy plural ISDEs.. 12 Hirofumi Osada, Tagged particle processes and their non-explosion criteria, Journal of the mathematical society of Japan, 10.2969/jmsj/06230867, 62, 3, 867-894, 2010.06. 13 Hirofumi Osada, Tomoyuki Shirai, Variance of the linear statistics of the Ginibre random point field, RIMS Kôkyûroku Bessatsu B6: Proceedings of RIMS Workshop on Stochastic Analysis and Applications eds. M. Fukushima and I. Shigekawa January, 2008 , B6, 2008.01. 14 Hirofumi Osada, Exotic Brownian motions, Kyushu Jarnal of Mathematics, 61巻1号, 2007.01. 15 Hirofumi Osada, Singular time changes of diffusions on Sierpinski carpets, Stochastic Process. Appl., １１６巻４号675--689, 2006.06. 16 Hirofumi Osada, Non-collision and collision properties of Dyson's model in infinite dimension and other stochastic dynamics whose equilibrium states are determinantal random point fields, Stochastic Analysis on Large Scale Interacting Systems, 39, 325-343, 2004, Dyson's model on interacting Brownian particles is a stochastic dynamics consisting of an infinite amount of particles moving in $\R$ with a logarithmic pair interaction potential. For this model we will prove that each pair of particles never collide. The equilibrium state of this dynamics is a determinantal random point field with the sine kernel. We prove for stochastic dynamics given by Dirichlet forms with determinantal random point fields as equilibrium states the particles never collide if the kernel of determining random point fields are locally Lipschitz continuous, and give examples of collision when H\"{o}lder continuous. In addition we construct infinite volume dynamics (a kind of infinite dimensional diffusions) whose equilibrium states are determinantal random point fields. The last result is partial in the sense that we simply construct a diffusion associated with the {\em maximal closable part} of {\em canonical} pre Dirichlet forms for given determinantal random point fields as equilibrium states. To prove the closability of canonical pre Dirichlet forms for given determinantal random point fields is still an open problem. We prove these dynamics are the strong resolvent limit of finite volume dynamics.. 17 Hirofumi Osada, Harnack inequalities for exotic brownian motions, Kyushu Journal of Mathematics, https://doi.org/10.2206/kyushujm.56.363, 56, 2, 363-380, 2002, Exotic Brownian motions are diffusion processes given by Dirichlet forms ℇ on L2 (S,μ), where the state space S is the support of a Radon measure μ in ℝd and the energy form ℇ is given by the integral of the canonical square field on ℝd with respect to μ. In most cases we take μ in such a way that μ is singular to the d-dimensional Lebesgue measure and, in particular, μ does not satisfy the doubling condition. The purpose of this paper is to prove the parabolic Harnack inequalities for these diffusions. We will do this with a refinement such that the dependence of μ, Poincaré and Sobolev constants is clarified. Because of the singularity of μ, diffusions are expected to behave in rather an unusual fashion. These Harnack inequalities will be used to show exotic properties of our diffusion processes in a forthcoming paper.. 18 Yuu Hariya, Hirofumi Osada, Diffusion processes on path spaces with interactions, Reviews in Mathematical Physics, https://doi.org/10.1142/S0129055X01000661, 13, 2, 199-220, 2001.02, We construct dynamics on path spaces C(ℝ;ℝ) and C([-r,r];ℝ) whose equilibrium states are Gibbs measures with free potential φ and interaction potential ψ. We do this by using the Dirichlet form theory under very mild conditions on the regularity of potentials. We take the carré du champ similar to the one of the Ornstein-Uhlenbeck process on C([0, ∞);ℝ). Our dynamics are non-Gaussian because we take Gibbs measures as reference measures. Typical examples of free potentials are double-well potentials and interaction potentials are convex functions. In this case the associated infinite-volume Gibbs measures are singular to any Gaussian measures on C(ℝ;ℝ).. 19 Hirofumi Osada, A family of diffusion processes on Sierpinski carpets, Probability Theory and Related Fields, https://doi.org/10.1007/PL00008761, 119, 2, 275-310, 2001.01, We construct a family of diffusions Pα = {Px.} on the d-dimensional Sierpinski carpet F̂. The parameter α ranges over dH < α < ∞, where dH = log(3d - 1)/log 3 is the Hausdorff dimension of the d-dimensional Sierpinski carpet F̂. These diffusions Pα are reversible with invariant measures μ = μ[α]. Here, μ are Radon measures whose topological supports are equal to F̂ and satisfy self-similarity in the sense that μ(3A) = 3α · μ(A) for all A ∈ ℬ(F̂). In addition, the diffusion is self-similar and invariant under local weak translations (cell translations) of the Sierpinski carpet. The transition density p = p(t, x, y) is locally uniformly positive and satisfies a global Gaussian upper bound. In spite of these well-behaved properties, the diffusions are different from Barlow-Bass' Brownian motions on the Sierpinski carpet.. 20 Hirofumi Osada, Tagged particles of interacting Brownian motions with skew symmetric drifts, Monte Carlo Methods and Applications, https://doi.org/10.1515/mcma.2001.7.3-4.339, 7, 3-4, 339-348, 2001.01, We prove an invariance principle for additive functionals of non reversible Markov processes with skew symmetric drifts.. 21 Hirofumi Osada, Herbert Spohn, Gibbs measures relative to Brownian motion, Annals of Probability, 27, 3, 1183-1207, 1999.07, We consider Brownian motion perturbed by the exponential of an action. The action is the sum of an external, one-body potential and a two-body interaction potential which depends only on the increments. Under suitable conditions on these potentials, we establish existence and uniqueness of the corresponding Gibbs measure. We also provide an example where uniqueness fails because of a slow decay in the interaction potential.. 22 Hirofumi Osada, Positivity of the self-diffusion matrix of interacting Brownian particles with hard core, Probability Theory and Related Fields, https://doi.org/10.1007/s004400050183, 112, 1, 53-90, 1998.01, We prove the positivity of the self-diffusion matrix of interacting Brownian particles with hard core when the dimension of the space is greater than or equal to 2. Here the self-diffusion matrix is a coefficient matrix of the diffusive limit of a tagged particle. We will do this for all activities, z > 0, of Gibbs measures; in particular, for large z - the case of high density particles. A typical example of such a particle system is an infinite amount of hard core Brownian balls.. 23 Hirofumi Osada, An invariance principle for Markov processes and Brownian particles with singular interaction, Annales de l'institut Henri Poincare (B) Probability and Statistics, https://doi.org/10.1016/S0246-0203(98)80031-9, 34, 2, 217-248, 1998, We prove an invariance principle for functionals of Markov processes. As an application we prove an invariance principle for tagged particles of Brownian particles with non-symmetric interactions.. 24 Hirofumi Osada, Interacting Brownian motions with measurable potentials, Proceedings of the Japan Academy Series A: Mathematical Sciences, https://doi.org/10.3792/pjaa.74.10, 74, 1, 10-12, 1998. 25 Hirofumi Osada, Dirichlet form approach to infinite-dimensional Wiener processes with singular interactions, Communications in Mathematical Physics, https://doi.org/10.1007/BF02099365, 176, 1, 117-131, 1996.01, We construct infinite-dimensional Wiener processes with interactions by constructing specific quasi-regular Dirichlet forms. Our assumptions are very mild; accordingly, our results can be applied to singular interactions such as hard core potentials, Lennard-Jones type potentials, and Dyson's model. We construct non-equilibrium dynamics.. 26 Hirofumi Osada, Long time estimates for transition probabilities of reflecting barrier Brownian motions, Probability theory and mathematical statistics (Tokyo, 1995), 396–402, 1996, We study long-time estimates for transition probabilities of reflecting-barrier Brownian motions in domains with infinitely many obstacles. We give a criterion for their being of the same order as a standard Brownian motion.. 27 Hirofumi Osada, Toshifumi Saitoh, An invariance principle for non-symmetric Markov processes and reflecting diffusions in random domains, Probability Theory and Related Fields, https://doi.org/10.1007/BF01192195, 101, 1, 45-63, 1995.03, We study an invariance principle for additive functionals of nonsymmetric Markov processes with singular mean forward velocities. We generalize results of Kipnis and Varadhan [KV] and De Masi et al. [De] in two directions: Markov processes are non-symmetric, and mean forward velocities are distributions. We study continuous time Markov processes. We use our result to homogenize non-symmetric reflecting diffusions in random domains.. 28 Hirofumi Osada, Self-similar diffusions on a class of infinitely ramified fractals, Journal of the Mathematical Society of Japan, https://doi.org/10.2969/jmsj/04740591, 47, 4, 591-616, 1995. 29 Hirofumi Osada, Estimates for transition probability of diffusion processes and their applicationstranslation of Sûgaku 41 (1989), no. 4, 335–344, Sugaku Expositions, 6, 1, 93-105, 1993. 30 Hirofumi Osada, Homogenization of reflecting barrier Brownian motions, Pitman Research Notes in Mathematics Series Asymptotic problems in probability theory: stochastic models and diffusions on fractals (Sanda/Kyoto, 1990), 283, 59–74, 1993, The author studies the case of a process X t which moves in a domain with stationary scatterers under certain geometric conditions (involving the isoperimetric constant of the domain). He proves that the limit of ϵX t/ϵ 2 is nondegenerate and gives an explicit lower bound for the determinant of its diffusion coefficient matrix.. 31 Hirofumi Osada, Cell fractals and equations of hitting probabilities, Probability theory and mathematical statistics (Kiev, 1991), 248-258, 1992. 32 Hirofumi Osada, Isoperimetric constants and estimates of heat kernels of pre Sierpinski carpets, Probability Theory and Related Fields, https://doi.org/10.1007/BF01198170, 86, 4, 469-490, 1990.12, The author calculated isoperimetric constants of the n-dimensional pre Sierpinski carpet Yn. As an application, he obtained the following estimate of the Neumann heat kernel pn (t, x, y) on Yn;[Figure not available: see fulltext.] where {Mathematical expression}. 33 Yoshikazu Giga, Tetsuro Miyakawa, Hirofumi Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity, Archive for Rational Mechanics and Analysis, https://doi.org/10.1007/BF00281355, 104, 3, 223-250, 1988.09. 34 Yoshikazu Giga, Tetsuro Miyakawa, Hirofumi Osada, Diffusion of vortices in planar Navier-Stokes flow, Sūrikaisekikenkyūsho Kōkyūroku Mathematical analysis of fluid and plasma dynamics, I (Kyoto, 1986), 656, 81-104, 1988. 35 Hirofumi Osada, Diffusion processes with generators of generalized divergence form, Kyoto Journal of Mathematics, 27, 4, 597–619, 1987. 36 Hirofumi Osada, Limit points of empirical distributions of vortices with small viscosity, The IMA Volumes in Mathematics and its Applications. Hydrodynamic behavior and interacting particle systems (Minneapolis, Minn., 1986), 9, 117–126, 1987, This paper, a sequel to the author's previous papers [J. Math. Kyoto Univ. 27 (1987), no. 4, 597–619; MR0916761; in Probabilistic methods in mathematical physics (Katata/Kyoto, 1985), 303–334, Academic Press, Boston, MA, 1987; MR0933829], is concerned with the vorticity equation which is equivalent to the two-dimensional nonstationary Navier-Stokes equations. The vorticity can be interpreted as the expectation of the McKean process associated with the vorticity equation. There is an n -particle system of stochastic differential equations which is expected to approximate the McKean process. The author proves that an n -particle system actually approximates the McKean process as n→∞ in the sense of distributions of processes. As a limit, a weak solution of the vorticity equation is constructed. In other words the author gives a rigorous derivation of the vorticity equation from an n -particle system as a propagation of chaos. No smallness assumptions on the Reynolds number are imposed. Since the velocity is determined by the vorticity through a convolution with a singular kernel, the stochastic equations involved in this paper have singularities. One should emphasize that the results do not follow from a general theory. This paper is a nice application of both analytic and probabilistic results in the above-mentioned papers.. 37 Hirofumi Osada, Propagation of chaos for the two-dimensional Navier-Stokes equation, Probabilistic methods in mathematical physics (Katata/Kyoto, 1985), 303-334, 1987, In this paper, the author studies the derivation of the vorticity formulation of two-dimensional Navier-Stokes equations from appropriate n -particle systems. In fact, the n -particle systems considered here are basically the ones corresponding to the vortex method. This derivation is completed under the assumption of large enough viscosity by probabilistic "propagation of chaos'' methods.. 38 Hirofumi Osada, Propagation of chaos for the two-dimensional Navier-Stokes equation, Proceedings of the Japan Academy Series A: Mathematical Sciences, 62, 1, 8-11, 1986, We establish a rigorous derivation of the two-dimensional vorticity equation associated with the Navier- Stokes equation from a many-particle system as a propagation of chaos.. 39 Hirofumi Osada, A stochastic differential equation arising from the vortex problem, Proceedings of the Japan Academy Series A: Mathematical Sciences, 61, 10, 333-336, 1985, The author studies the system of stochastic differential equations dz i =∑ j≠i α j K(z i −z j )dt+σdW i ,i=1,⋯,n , where W i are 2-dimensional independent Brownian motions, K(z)≡K(x,y)=(G y ,−G x ) , and G(z)=−(2π) −1 log(|z|) . The 2-dimensional variables z i represent the positions of n vortices in a viscous and incompressible fluid, with vorticity intensities α i , respectively. The constant σ is related to the viscosity. The drift is singular on a manifold S in R 2n . It is shown that the first-passage time to S is infinite with probability 1.. 40 Hirofumi Osada, Moment estimates for parabolic equations in the divergence form., Kyoto Journal of Mathematics, 25, 3, 473-488, 1985, The author proves certain generalizations of the Nash estimate [ J. Nash , Amer. J. Math. 80 (1958), 931–954; MR0100158]. Let p(s,x,t,y) be a measurable fundamental solution of ∇ t −A≡∇ t −∑ n i,j=1 ∇ i a ij ∇ j . It is assumed that ∑ n i,j=1 a ij (t,x)ξ i ξ j ≥ν|ξ| 2 and that there exist a symmetric matrix b ij (t) and a matrix c ij (t,x) , such that a ij (t,x)=b ij (t)+c ij (t,x) , and, moreover, sup 1≤j≤n ∑ n i=1 |b ij (t)|≤λ and sup 1≤i,j≤n |c ij (t,x)|≤μ/n for every (t,x)∈[0,∞)×R n . It is also assumed that the constants λ , μ , ν are independent of n . Under these assumptions the author proves the following: If m,q are nonnegative integers, then there exist positive constants C 1 , C 2 , dependent only on λ , μ , ν , m and q , such that C 1 n q+1 (t−s) (m+q)/2 ≤∫ R n (∑ i=1 n |x i −y i | m )(∑ i=1 n |x i −y i |) q p(s,x,t,y)dy≤C 2 n q+1 (t−s) (m+q)/2 for all x∈R n , 0≤s The estimates of this work can be used in the solution of the problem of interacting diffusion processes in probability theory, as formulated by H. P. McKean, Jr. [Stochastic differential equations (Washington, D.C., 1967), 41–57, Air Force Office Sci. Res., Arlington, Va., 1967; MR0233437].. 41 Hirofumi Osada, Shinichi Kotani, Propagation of chaos for the Burgers equation, Journal of the Mathematical Society of Japan, https://doi.org/10.2969/jmsj/03720275, 37, 2, 276-294, 1985. 42 Hirofumi Osada, Homogenization of diffusion processes with random stationary coefficients, Lecture Notes in Mathematics Probability theory and mathematical statistics (Tbilisi, 1982),, 1021, 507-517, 1983, 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) ..