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 $|x-y|<sup>{-d-2}$. As the associated jumping kernel fails to be $L<sup>2$-integrable yet admits a finite $α$-th moment for all $α\in (0,2)$, we refer to the corresponding process $(X<sup>\w_t)_{t\ge0}$ 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 $α$-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 $$(\log k)<sup>{-\frac{1}{2}+\frac{1}{2(d-2)}+\varepsilon}$$ with any $\varepsilon&gt;0$ for all $d&gt;3$.

• The paper establishes that rescaled long-range random walks converge to Brownian motion in dimensions d>3 under a critical jump index.
• The paper introduces localized correctors and multi-scale Poincaré inequalities to overcome the divergence of second moments in the critical regime.
• The paper demonstrates logarithmically slow convergence rates, highlighting the transition between nonlocal interactions and classical diffusive behavior.

Quantitative Stochastic Homogenization for Long-Range Random Walks with Critical Jump Index

Introduction and Theoretical Framework

This paper addresses the quantitative stochastic homogenization problem for symmetric random walks on $\mathbb{Z}^d$ with long-range jumps characterized by a critical decay in the transition kernel; specifically, the probability of a jump from $x$ to $y$ is proportional to $|x-y|^{-d-2}$. The model is situated within the random conductance paradigm, where a collection of i.i.d. conductance weights $\{w_{x,y}\}$ defines the heterogeneous random environment.

The criticality of the jump index manifests in the delicate balance between finite $\alpha$-th moments for all $\alpha \in (0,2)$ and the divergence of the second moment. Consequently, the associated random walk, under a specifically calibrated space-time scaling (neither classical diffusive nor $\alpha$-stable), exhibits homogenization to Brownian motion. This critical regime provides a challenging and nuanced setting for quantitative analysis, diverging from both pure local (nearest neighbor) and genuinely nonlocal ($\alpha$-stable, $\alpha\in(0,2)$) settings.

Main Results

The central result demonstrates that, under ergodic and uniform ellipticity assumptions on the environment (Assumption H1), the space-time rescaled process converges in distribution to a Brownian motion in dimension $x$0. Specifically, the generator of the random walk,

$x$1

operates in an environment where the jump kernel is at the critical decay threshold.

The scaling for homogenization is determined to be $x$2, reflecting slower-than-diffusive, yet faster than stable, behavior. The main quantitative homogenization theorem establishes that, for the $x$3-resolvent $x$4 of the scaled generator,

$x$5

holds with high probability for all sufficiently large $x$6, any test function $x$7 in a suitable class, and any $x$8. Here, $x$9 is the resolvent of the effective operator $y$0, with

$y$1

Methodological Advances and Technical Ingredients

Localized Corrector Construction

Unlike the nearest-neighbor or subcritical $y$2-stable cases where a global $y$3-corrector may exist and is central to homogenization analysis, the non-integrability of the jump kernel at the critical index obstructs such constructions. Instead, the authors construct localized correctors within finite regions, whose $y$4-energies diverge algebraically as the domain grows. The speed of divergence is carefully quantified and shown to play a decisive role in setting the rate of convergence in homogenization.

Poincaré-Type Inequalities

The paper develops both a local weak-type and a multi-scale Poincaré inequality adapted to the Dirichlet form determined by the random environment. The weak-type version reflects the nonlocality and long-range decay of the jump kernel, reflected in rates containing factors of $y$5. Poincaré inequalities at multiple scales facilitate a robust comparison across scales and underpin the averaging required for quantitative bounds.

Discrete-to-Continuum Analysis and Taylor Expansions

To establish quantitative rates, the discrete generators are carefully compared with the limiting Laplacian using multiscale Taylor expansions and judicious truncations. Crucially, the third-order Taylor terms and truncation errors are shown to dominate the error analysis due to the critical nature of the kernel.

The comparison is realized not only at the level of the generator but also for the resolvent. Deviations arising from the random environment are isolated and controlled via concentration inequalities (e.g., Hoeffding-type inequalities for sums of i.i.d. bounded variables modulated by test functions and their derivatives), and the scaling of variances is carefully estimated.

Strong Numerical and Theoretical Claims

• Scaling Regime: The authors rigorously prove that the scaling order $y$6 is the appropriate normalization for homogenization at the critical jump index. This is strictly distinct from both purely diffusive ($y$7) and subdiffusive ($y$8, $y$9) regimes.
• Homogenized Limit Operator: The limit is proven to be a constant coefficient Laplacian, $|x-y|^{-d-2}$0, where $|x-y|^{-d-2}$1 only depends on the mean of the random conductances, unlike the nearest-neighbor case where the homogenized constant depends on the (localized) corrector.
• Convergence Rate: The $|x-y|^{-d-2}$2-convergence rate for the scaled resolvent is given by $|x-y|^{-d-2}$3 up to an arbitrarily small $|x-y|^{-d-2}$4, demonstrating logarithmically slow homogenization compared to the polynomial rates in subcritical cases. Notably, the rate is independent of the detailed law of the conductances, assuming only uniform ellipticity and ergodicity.

Discussion, Implications, and Future Directions

Practical and Theoretical Implications

The results illuminate the delicate transition between anomalous and normal diffusion in stochastic media with long-range jumps. In heterogeneous random media relevant to physical models of anomalous transport, disordered systems, and nonlocal PDEs, understanding rates of convergence to the homogenized limit is vital for both simulations and theoretical predictions.

From a mathematical perspective, this work bridges the gap between discrete-space nonlocal interactions and continuum, local (differential) operators in the critical scaling regime, providing techniques likely transferable to other models exhibiting critical behavior with diverging moments.

Potential Extensions

• Lower Dimensions and Critical Thresholds: The analysis here covers $|x-y|^{-d-2}$5; exploring sharp convergence rates or even qualitative homogenization for $|x-y|^{-d-2}$6 (where logarithmic divergences become more severe) or $|x-y|^{-d-2}$7 would be natural extensions.
• Degenerate Environments: The assumption of uniform ellipticity could be further relaxed to accommodate degenerate or heavy-tailed random conductances, which are common in physical applications.
• Time-Dependent and Dynamical Environments: Methodological components (localized correctors, multiscale estimates) appear adaptable to broader classes of non-stationary or time-dependent random environments.
• Other Scaling Limits: The approach opens room for investigating more complex multi-scale or mixed local-nonlocal models, including those driven by fractional Laplacians perturbed by rapidly oscillating random coefficients.

Conclusion

This paper provides a comprehensive quantitative homogenization theory for random conductance models with critical long-range jumps, establishing slow logarithmic convergence rates and delineating the precise scaling regime in higher dimensions. The systematic construction of localized correctors, the development of adapted Poincaré inequalities, and the sharp error estimates may provide a methodological template for future work in nonlocal stochastic homogenization and for broader random media models.

Citation: "Quantitative stochastic homogenization for long-range random walks with critical jump index" (2604.21162).