A note on the stochastic weakly* almost periodic homogenization of fully nonlinear elliptic equations

摘要

A function f is an element of BUC(R-d) is said to be weakly* almost periodic, denoted f is an element of W*AP(R-d), if there is g is an element of AP(R-d), such that, M(vertical bar f - g vertical bar) = 0, where BUC(R-d) and AP(R-d) are, respectively, the space of bounded uniformly continuous functions and the space of almost periodic functions, in R-d, and M(h) denotes the mean value of h, if it exists. We give a very simple direct proof of the stochastic homogenization property of the Dirichlet problem for fully nonlinear uniformly elliptic equations of the form F(omega, x/epsilon, D(2)u) = 0, x is an element of U, in a bounded domain U subset of R-d, in the case where for almost all omega is an element of Omega, the realization F (omega, epsilon, M) is a weakly* almost periodic function, for all M is an element of S-d, where S-d is the space of d x d symmetric matrices. Here (Omega, mu, F) is a probability space with probability measure mu and sigma-algebra F of mu-measurable subsets of Omega. For each fixed M is an element of S-d, F(omega, y, M) is a stationary process, that is, F(omega, y, M) = (F) over tilde (T(y)omega, M) := F(T(y)omega, 0, M) , where T(y) : Omega -> Omega is an ergodic group of measure preserving mappings such that the mapping (omega, y) -> T(y)omega is measurable. Also, F(omega, y, M), M is an element of S-d, is uniformly elliptic, with ellipticity constants 0 < lambda < Lambda independent of (omega, y) is an element of Omega x R-d. The result presented here is a particular instance of the general theorem of Caffarelli, Souganidis and Wang, in CPAM 2005. Our point here is just to show a straightforward proof for this special case, which serves as a motivation for that general theorem, whose proof involves much more intricate arguments. We remark that any continuous stationary process verifies the property that almost all realizations belong to an ergodic algebra on R-d, and that W*AP(R-d) contains all the ergodic algebras on R-d so far known.