An ergodic theorem with weights and applications to random measures, RW homogenization and IPS hydrodynamics on point processes

Abstract

We prove a multidimensional ergodic theorem with weighted averages for the action of the group ${\mathbb{Z}}^d$ on a probability space. At level $n$ weights are of the form $n^{-d}\ddot{I}ˆ(j/n)$, $j\in {\mathbb{Z}}^d$, for real functions $\ddot{I}ˆ$ decaying suitably fast. We discuss applications to random measures and to quenched stochastic homogenization of random walks on simple point processes with long-range random jump rates, allowing to remove the technical Assumption (A9) from \cite[Theorem~4.4]{Fhom1}. This last result concerns also some semigroup and resolvent convergence particularly relevant for the derivation of the quenched hydrodynamic limit of interacting particle systems via homogenization and duality. As a consequence we show that also the quenched hydrodynamic limit of the symmetric simple exclusion process on point processes stated in \cite[Theorem~4.1]{F_SEP} remains valid when removing the above mentioned Assumption (A9).

To appear in the special issue of SPA in memory of Francis Comets. 35 pages, 2 figures. The presentation is now self-contained and with additional examples