Quantitative stochastic homogenization for long-range random walks with critical jump index

Abstract

In this paper, we study the stochastic homogenization for a class of symmetric random walks in random conductance model, whose one-step transition probability from $x$ to $y$ is proportional to $|x-y{|}^{-d-2}$. As the associated jumping kernel fails to be $L^2$-integrable yet admits a finite $\hat{I}\pm$-th moment for all $\hat{I}\pm \in (0,2)$, we refer to the corresponding process $(X^\w_t)_{t\ge0}$ as a long-range random walk with critical jump index. In this critical regime, the scaled process $(k^{-1}{X}_{k^2(\log k{)}^{-1}t}{)}_{t\ge 0}$, whose scaling order is different from the diffusive scaling and the $\hat{I}\pm$-stable scaling, converges to a Brownian motion. Besides characterizing the limiting Brownian motion, we will give a convergence rate for associated scaled resolvents, which obeys the order $(\log k{)}^{-\frac{1}{2}+\frac{1}{2(d-2)}+\epsilon }$ with any $\epsilon >0$ for all $d>3$.

23 pages