Green’s function for elliptic systems: Existence and stochastic bounds

We study the Green function G associated to the operator −∇ · a∇ in Rd, when a = a(x) is a (measurable) bounded and uniformly elliptic coefficient field. An example of De Giorgi implies that, in the case of systems, the existence of a Green’s function is not ensured by such a wide class of coefficient fields a. We give a more general definition of G and show that for every bounded and uniformly elliptic a, such G exists and is unique. In addition, given a stationary ensemble $\langle\cdot\rangle$ on a, we prove optimal decay estimates for $\langle|G|\rangle $ and $\langle|∇G|\rangle$. Under assumptions of quantification of ergodicity for $\langle\cdot\rangle$, we extend these bounds also to higher moments in probability. These results play an important role in the context of quantitative stochastic homogenization for −∇ · a∇. This talk is based on joint works with Peter Bella, Joseph Conlon and Felix Otto.