Ergodic theory and application to nonconvex homogenization
Let \(\{\tau_ z: z\in S= k\mathbb{Z}^ d\}\) be a group of \(P\)-preserving transformations in a probability space \((\Sigma, T, P)\). Let \(\mu\) be a random Borel measure, mapping \(\Sigma\) into the set \(M(\mathbb{R}^ d)\) of nonnegative regular Borel measures on \(\mathbb{R}^ d\), and satisfying \(\mu(\tau_ z \omega)(A)= \mu(\omega)(z+ A)\). For a sequence \(\varepsilon_ n> 0\) tending to 0, put \[ \mu_ n(\omega)(A)= \varepsilon^ d_ n \mu(\omega)(A/\varepsilon_ n), \] where \(A\) is a Borel subset of a bounded open set \(\Omega\subset \mathbb{R}^ d\). The main result asserts: If \(\mu_ n\) is almost surely tight, then almost surely \(\mu_ n(\omega)\) converges in the ``narrow'' topology. (Narrow remains undefined, but seems to refer to the vague topology.) The limit is identified. There are applications to stochastic homogenization of nonconvex integral functionals.
scientific article; zbMATH DE number 627467