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

TL;DR

This paper investigates the stochastic homogenization of long-range symmetric random walks with a critical jump index, demonstrating convergence to Brownian motion and providing explicit convergence rates for scaled resolvents.

Contribution

It characterizes the limiting behavior of long-range random walks with critical jump index and establishes convergence rates for scaled resolvents in this regime.

Findings

Scaled process converges to Brownian motion with a non-standard scaling.

Provides explicit convergence rate for scaled resolvents involving logarithmic factors.

Identifies the critical regime where the jump kernel's integrability properties influence the limit.

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 $\alpha$-th moment for all $\alpha\in (0,2)$, 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 $\bigl(k^{-1}X_{k^2(\log k)^{-1}t}\bigr)_{t\ge 0}$, whose scaling order is different from the diffusive scaling and the $\alpha$-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 $(\log k)^{-\frac{1}{2}+\frac{1}{2(d-2)}+\varepsilon}$ with any $\varepsilon>0$ for all $d>3$.