Analysis Seminar

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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).