8. Introduction to Stochastic Homogenization

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.