Estimates in probability of the residual between the random and the homogenized solutions of one‐dimensional second‐order operator
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The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one-dimensional second-order elliptic operators with random coefficients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coefficient. In the mathematical literature there are now many papers devoted to homogenization of random op- erators with coefficients being stationary random field (see, for instance, (3) and references therein) and of operators posed in randomly perforated domain (see (2,3)). But all these results are mainly giving the convergence of the solutions towards the solution of the limit (or homogenized) equation, without estimate of the residual. The first successful attempt to give such an estimate is the work of Yurinski (6), where the expectation of some norm of the residual for the divergence form second-order elliptic random operator is estimated by a positive power of a small parameter " that characterizes the microscopic length scale. This power of " depends only on the dimension of the space, the ellipticity constant and on some characteristics of the mixing conditions; but this power is implicit and could not be computed explicitly. Later, similar problems have been studied for symmetric elliptic systems (5); in this case the residual is estimated by some negative power ofj log"j, which could not, once more, be computed explicitly. The aim of our paper is to investigate in the one-dimensional case the probabilistic property of the residual. We assume that the coefficients of the operator is a stationary random field satisfying strong or uniform mixing condition. The first two sections deal with the case when the corresponding mixing coefficient decays like a negative power of a distance. Namely, in the first part, we suppose that >1, i.e., that the random variables a(x, )a nda(x +d,) are weakly dependent for large d. This allows to apply the central limit theorem. In the second part we study the case when the mixing properties of the random field are not so good, i.e., when 6 1. Finally, in the third part, using large deviation type estimates and assuming that the mixing coefficient decays like the exponent of some power of the distance, we get more precise bounds in probability for the fields with such "good" mixing properties. It should be noted that
[1] M. Freidlin,et al. Random Perturbations of Dynamical Systems , 1984 .
[2] Alʹbert Nikolaevich Shiri︠a︡ev,et al. Theory of martingales , 1989 .
[3] ON THE ERROR OF AVERAGING SYMMETRIC ELLIPTIC SYSTEMS , 1990 .
[4] V. V. Yurinskii. Averaging of symmetric diffusion in random medium , 1986 .
[5] U. Hornung. Homogenization and porous media , 1996 .