Quantitative stochastic homogenization for random conductance models with stable-like jumps

TL;DR

This paper develops a quantitative framework for stochastic homogenization of long-range jump random conductance models on integer lattices, providing explicit polynomial convergence rates with logarithmic adjustments.

Contribution

It introduces a novel quantitative homogenization approach for stable-like jump models with explicit convergence rates.

Findings

Established polynomial rates of convergence for homogenization.

Derived explicit bounds with logarithmic corrections.

Extended homogenization theory to long-range jump processes.

Abstract

We consider random conductance models with long range jumps on $\Z^d$, where the one-step transition probability from $x$ to $y$ is proportional to $w_{x,y}|x-y|^{-d-\alpha}$ with $\alpha\in (0,2)$. Assume that $\{w_{x,y}\}_{(x,y)\in E}$ are independent, identically distributed and uniformly bounded non-negative random variables with $\Ee w_{x,y}=1$, where $E$ is the set of all unordered pairs on $\Z^d$. We obtain a quantitative version of stochastic homogenization for these random walks, with explicit polynomial rates up to logarithmic corrections.