Gaussian fluctuation in stochastic homogenization and the random conductance model

Jim Nolen, Duke
Fine Hall 214

This talk is about solutions to a linear, divergence-form elliptic PDE (or a discrete variant) with a conductivity coefficient that varies randomly with respect to the spatial variable.  Although the PDE is linear, its solution depends on the random coefficients in a non-local and non-linear way.  It is well-known that the random solutions to the PDE may exhibit a homogenization phenomenon at large scales.  If we think of this as a kind of law of large numbers for the solution, then it is natural to ask about fluctuations and large deviations. Gloria and Otto have derived bounds on moments of the solution which scale in an optimal way.  By combining these bounds with Stein's method of normal approximation, we prove a quantitative central limit theorem for functionals of the solution. The talk is based on some joint works with Antoine Gloria, Jean-Christophe Mourrat, and Felix Otto.