Stochastic homogenization of random walks on point processes

TL;DR

This paper proves homogenization results for random walks on random point processes, showing convergence to Brownian motion under broad conditions without ellipticity assumptions, applicable to various models like conductance and percolation.

Contribution

It introduces a novel combination of two-scale convergence and Palm theory to establish homogenization for random walks on atomic measures without ellipticity constraints.

Findings

Homogenization of the associated Markov generators to Brownian motion.

Quenched convergence of the $L^2$-Markov semigroup and resolvent.

Applicability to models like random conductance, Mott hopping, and percolation clusters.

Abstract

We consider random walks on the support of a random purely atomic measure on $\mathbb{R}^d$ with random jump probability rates. The jump range can be unbounded. The purely atomic measure is reversible for the random walk and stationary for the action of the group $\mathbb{G}=\mathbb{R}^d$ or $\mathbb{G}=\mathbb{Z}^d$. By combining two-scale convergence and Palm theory for $\mathbb{G}$-stationary random measures and by developing a cut-off procedure, under suitable second moment conditions we prove for almost all environments the homogenization for the massive Poisson equation of the associated Markov generators. In addition, we obtain the quenched convergence of the $L^2$-Markov semigroup and resolvent of the diffusively rescaled random walk to the corresponding ones of the Brownian motion with covariance matrix $2D$. For symmetric jump rates, the above convergence plays a crucial role in the derivation of hydrodynamic limits when considering multiple random walks with site-exclusion or zero range interaction. We do not require any ellipticity assumption, neither non-degeneracy of the homogenized matrix $D$. Our results cover a large family of models, including e.g. random conductance models on $\mathbb{Z}^d$ and on general lattices (possibly with long conductances), Mott variable range hopping, simple random walks on Delaunay triangulations, simple random walks on supercritical percolation clusters.