An ergodic theorem with weights and applications to random measures, homogenization and hydrodynamics

We prove a multidimensional ergodic theorem with weighted averages for the action of the group $Z^d$ on a probability space. At level $n$ weights are of the form $n^{-d}\psi (j/n)$, $j\in Z^d$, for real functions $\psi$ 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 [Faggionato 2023, Theorem 4.4]. 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 [Faggionato 2022, Theorem 4.1] remains valid when removing the above mentioned Assumption (A9).