Gaussian fluctuation in stochastic homogenization and the random conductance model
Gaussian fluctuation in stochastic homogenization and the random conductance model
This talk is about solutions to a linear, divergence-form ellipticÌýPDE (or a discrete variant) with a conductivity coefficient that variesÌýrandomly with respect to the spatial variable.Ìý Although the PDE is linear,Ìýits solution depends on the random coefficients in a non-local andÌýnon-linear way.Ìý It is well-known that the random solutions to the PDE mayÌýexhibit a homogenization phenomenon at large scales.Ìý If we think of this asÌýa kind of law of large numbers for the solution, then it is natural to askÌýabout fluctuations and large deviations. Gloria and Otto have derived boundsÌýon moments of the solution which scale in an optimal way.Ìý By combiningÌýthese bounds with Stein's method of normal approximation, we prove aÌýquantitative central limit theorem for functionals of the solution. The talkÌýis based on some joint works with Antoine Gloria, Jean-Christophe Mourrat,Ìýand Felix Otto.