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 Z𝑑 on a probability space. At level 𝑛 weights are of the form 𝑛−𝑑𝜓(𝑗∕𝑛), 𝑗 ∈ Z𝑑 , for real functions 𝜓 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).