# Analysis Seminar

#### Quantitative Central Limit Theorem for the Effective Diffusion in iid Environments

Speaker: Antoine Gloria

Location: Warren Weaver Hall 1302

Date: Thursday, May 1, 2014, 11 a.m.

Synopsis:

In this talk I shall present a quantitative central limit theorem for the approximation $$A_L$$ of the effective diffusion $$A_{hom}$$ by periodization (on the $$L$$-torus) in a discrete iid environment in dimension $$d\geq 2$$. On the one hand, using Stein's method, I shall prove a quantitative estimate on the decay of the Wasserstein distance between $$A_L$$ minus its expectation rescaled by the square-root of its variance and a normal random variable. On the other hand, I shall prove that the rescaled variance $$\sigma_L2 := L^d \mathrm{var}(A_L)$$ has a limit $$\sigma2$$ as $$L$$ goes to infinity, and shall quantify in terms of $$L$$ the convergence of the expectation of $$A_L$$ to $$A_{hom}$$ and the convergence of $$\sigma_L2$$ to $$\sigma2$$. The proofs of these results are based on a seriesof works in collaboration with Marahrens, Mourrat, Neukamm, and Otto. Combining both types of results then allows us to bound the Wasserstein distance between $$\sigma^{-1} L^{d/2} (A_L-A_{hom})$$ and a normal random variable by a constant times $$L^{-d/2}\log^d L$$, which we think is optimal. This is joint work with Jim Nolen (Duke).