Scott N. Armstrong: Introductory program: Quantitative stochastic homogenization and quenched higher regularity for nonlinear elliptic equations

I will present a strategy for obtaining precise quantitative behavior in stochastic homogenization for elliptic equations: both linear and nonlinear, in both divergence and nondivergence forms. The main issue is to obtain "quenched higher regularity" for solutions, showing they are more regular than solutions of general equations with measurable coefficients (we need Lipschitz estimates for divergence form equations and C^1,1 estimates for nondivergence form equations). We show how to prove such estimates using subadditivity methods. Combining these estimates with concentration inequalities, in both cases we obtain optimal bounds on the approximate cell problem in every dimension and characterize in exactly which dimensions exact stationary correctors exist. This extends the work of Gloria-Marahrens-Neukamm-Otto to the nonlinear (variational) setting and is completely new in the nondivergence form case (even in the linear setting). Some of the results in the talk are joint with Charles Smart and others with Antoine Gloria.