Title: Quantitative homogenization of random environments

Abstract:

If we observe a random walk moving among random i.i.d. conductances on larger and larger diffusive scales, it is known that we obtain a Brownian motion with a constant covariance matrix in the limit. I will describe recent progress on the question of quantifying this convergence. From the PDE's perspective, one considers instead the (divergence form) generator of the random walk, and similarly if one looks at this operator from a distance, it tends to look like an operator with constant "homogenized" coefficients. An important question in numerical analysis is to be able to compute these coefficients. We will see how the analysis of certain techniques of estimation of these coefficients can be linked with the probabilistic problem.

References:

- Variance decay for functionals of the environment viewed by the particle, Ann. Inst. Henri Poincaré Probab. Stat. 47 (1), 294-327 (2011).

- Other refences here.