5 May 2015
Stochastic homogenization shows that solutions to an elliptic problem with rapidly oscillating, ergodic random coefficients can be effectively described by an elliptic problem with homogeneous, deterministic coefficients. The definition of the latter is based on the construction of a "corrector" and invokes an elliptic operator that acts on the probability space of admissible coefficient fields. While qualitative homogenization is well understood and classical, quantitative results (e.g. estimates on the homogenization error and approximations to the homogenized coefficients) have only been obtained recently. In the talk we discuss an optimal estimate on the associated semigroup (usually called the "random walk in the random environment") and show that it decays with an algebraic rate. The result relies on a link between a Spectral Gap of a Glauber dynamics on the space of coefficient fields (a notion that we borrow from statistical mechanics) and heat kernel estimates. As applications we obtain moment bounds on the corrector and an optimal convergence rate for the approximation of the homogenized coefficients via periodic representative volume elements.
- Functional Analysis Seminar