This chapter introduces the reader to stochastic homogenization problems describing processes in heterogeneous media whose microstructure is not periodic and, moreover, cannot be described with certainty. As in the previous chapters we stick to the case study conductivity problem but assume no periodicity of the coefficients. Instead we consider the case when the coefficients are rapidly oscillating random fields. We present a detailed proof of the classical theorem on existence of the homogenized limit for problems with stationary and ergodic random coefficients. Before proving this theorem we introduce the reader to stochastic models of heterogeneous media. This is done using basic examples such as the random checkerboard and the Poisson cloud. These examples are easy on the intuitive level but their rigorous mathematical understanding requires significant effort for someone with no experience in stochastic modeling. We conclude the chapter by a brief discussion of recent developments and provide a number of references for further reading.
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Armstrong, S., Caraliaguet P.: Stochastic homogenization of quasilinear Hamilton-Jacobi equations and geometric motions J. Eur. Math. Soc., to appear
Armstrong, S., Smart, C.: Quantitative stochastic homogenization of convex integral functionals. Ann. Sci. Éc. Norm. Supér. (4)
49(2), 423–481 (2016)
Berlyand, L. and Mityushev, V.: Increase and decrease of the effective conductivity of two phase composites due to polydispersity. J. Stat. Phys.
118(3/4), 481–509 (2005)
Blanc, X., Le Bris, C., Legoll, F.: Some variance reduction methods for numerical stochastic homogenization. Philos. Trans. Roy. Soc. A
374 (2066), 20150168 (2016)
Bourgeat, A., Piatnitski, A.: Estimates in probability of the residual between the random and the homogenized solutions of one-dimensional second-order operator. Asymptot. Anal.
21(3–4), 303–315 (1999)
Bourgeat, A., Piatnitski, A.: Approximations of effective coefficients in stochastic homogenization. Ann. Inst. H. Poincaré
40(2), 153–165 (2004)
Caffarelli, L.A., Souganidis, P.E.: Rates of convergence for the homogenization of fully nonlinear uniformly elliptic PDE in random media. Invent. Math.
180(2), 301–360 (2010)
Caffarelli, L.A., Souganidis, P.E., Wang, L.: Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media. Comm. Pure Appl. Math.
58(3), 319–361 (2005)
Doob, J.L.: Stochastic Processes. John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London (1953)
Gloria, A., Mourrat, J.-C.: Spectral measure and approximation of homogenized coefficients. Probab. Theory Related Fields
154 (1–2), 287–326 (2012)
Gloria, A., Otto, F.: An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab.
39(3), 779–856 (2011)
Gloria, A., Neukamm, S., Otto, F.: Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics. Invent. Math.
199(2), 455–515 (2015)
Jikov, V. V., Kozlov, S. M., Oleinik, O. A.: Homogenization of Differential Operators and Integral Functionals. Springer-Verlag (1994)
Shiryaev, A. N.: Probability (volume 95 of Graduate texts in mathematics). Springer-Verlag New York (1996)
Sinai, Y. G.: Probability Theory: an Introductory Course. Springer Science & Business Media (2013)
Yurinskiı̆, V. V.: Averaging of symmetric diffusion in a random medium. Sibirsk. Mat. Zh.
27(4), 167–180, 215 (1986)