Quantitative stochastic homogenization via Malliavin calculus
Abstract: This talk is about stochastic homogenization of linear elliptic equations in divergence form. Let $a(x)=h(G(x))$ be a diffusion coefficient field, where $h$ is a Lipschitz function and $G$ is a Gaussian field (with possibly thick tail). Solutions $u_\varepsilon$ of elliptic equations $-\nabla \cdot a(\cdot/\varepsilon) \nabla u_\varepsilon = \nabla \cdot f$ in $\mathbb R^d$ with such random heterogeneous coefficients $a$ both oscillate spatially and fluctuate randomly at scale $\varepsilon >0$. I will show how suitable quantitative two-scale expansions allow one to reduce the analysis of oscillations and fluctuations of solutions to bounds on the corrector and fluctuations of the homogenization commutator, respectively. The main probabilistic ingredient is Malliavin calculus, and the main analytical ingredient is large-scale elliptic regularity. This is based on joint works with Mitia Duerinckx, Julian Fischer, Stefan Neukamm, and Felix Otto.