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

2023/12/19 by Faggionato, A.
#35B27 #37A30 #60G55 #60K37 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2312.12348

Abstract

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 ψ(j/n), j∈ ℤ^d, 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 \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).

Related

• Stochastic homogenization of random walks on point processes
• From quenched invariance principle to semigroup convergence with applications to exclusion processes
• Local ergodicity in the exclusion process on an infinite weighted graph
• Hydrodynamic limit of simple exclusion processes in symmetric random environments via duality and homogenization
• Semi-deterministic processes with applications in random billiards