Quantitative stochastic homogenization - a semigroup approach

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.