Quantitative stochastic homogenization - a semigroup approach

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