Stochastic homogenization for diffusions in random media

This thesis deals with stochastic homogenization of Markov diffusion processes associated to second order uniformly elliptic partial differential operators in divergence form with random coefficients. Exploiting the beautiful relationship between diffusions, infinitesimal generators, transition functions and parabolic Green functions, thus employing tools from both probability theory and PDEs, it is possible to prove a quenched invariance principle for symmetric diffusions in random media. When the environment is stationary and ergodic, this principle states the convergence in law of a diffusion process under diffusive scaling to a deterministic multiple of the Brownian motion for almost all realizations of the random coefficients. In order to prove it, we proceed with a mostly analytic approach which bypasses the need for the “environment viewed from the particle”, based on recent developments by Scott Armstrong, Tuomo Kuusi and Jean-Christophe Mourrat.