HOMOGENIZATION IN A RANDOM LATTICE - ANDERSON MELCHOR HERNANDEZ - UNIVERSITÀ DI PAVIA
The behavior of many heterogeneous media, such as porous or composite materials, is described by partial differential equations with coefficients that randomly vary on a small scale. At the macroscopic scale (large compared to the dimension of the heterogeneities) such media often show an effective behavior, which is typically deterministic and emerges as the averaging of the stochastic structure at the microscopic small-scale. The effect of this averaging process, called homogenization, is the independence, of the coefficients of the effective macroscopic model, of the spatial variable. In a pioneering work, Kozlov, Papanicolaou, and Varadhan studied (steady) heat conduction in a randomly inhomogeneous conducting medium and obtained a qualitative homogenization result for stationary, ergodic conductivities. The aim of this talk is to describe the asymptotic behavior of an elliptic operator in a lattice depending on a sequence of independent random variables. We will use a two-scale expansion combined with a spectral gap inequality to predict the rate of convergence towards the limit homogenized operator. Moreover, we show the corresponding quantitative central limit theorem to the random walk evolving in a random lattice.
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