20 November 2015
Heterogeneous media, like a sediment, are often naturally described in statistical terms. How to extract their effective behaviour on large scales, like the permeability in Darcy's law, from the statistical specifications? A practioners numerical approach is to sample the medium according to these specifications and to determine the permeability in the Cartesian directions by imposing simple boundary conditions. What is the error made in terms of the size of this "representative volume element''? Our interest in what is called "stochastic homogenization'' grew out of this error analysis.
In the course of developing such an error analysis, connections with the classical regularity theory for elliptic operators have emerged. It turns out that the randomness, in conjunction with statistical homogeneity, of the coefficient field (which can be seen as a Riemannian metric) generates large-scale regularity of harmonic functions (w.r.t. the corresponding Laplace-Beltrami operator). This is embodied by a hierarchy of Liouville properties:
Almost surely, the space of harmonic functions of given but arbitrary growth rate has the same dimension as in the flat (i. e. Euclidean) case.
Classical examples show that from a deterministic point of view, the Liouville property fails already for a small growth rate:
There are (smooth) coefficient fields, which correspond to the geometry of a cone at infinity, that allow for sublinearly growing but non-constant harmonic functions.