Periodic homogenization of convolution type operators with heavy tails
Andrey Piatnitski, Elena Zhizhina
TL;DR
The paper investigates periodic homogenization for nonlocal convolution operators with heavy-tailed kernels in the domain of attraction of a symmetric $\alpha$-stable law, modulated by a periodic coefficient. Using compactness arguments rather than correctors, it proves strong resolvent convergence of $(-L^{\varepsilon}+m)^{-1}$ in $L^2(\mathbb{R}^d)$ to the resolvent of an effective nonlocal operator $L^{\rm eff}$ whose kernel is $\Lambda^{\rm eff}(x,y)=\overline{\Lambda} \; k(\frac{x-y}{|x-y|})$ and is given by $L^{\rm eff}u(x)=\mathrm{p.v.}\int\frac{\overline{\Lambda} k(\frac{x-y}{|x-y|})(u(y)-u(x))}{|x-y|^{d+\alpha}}\,dy$. The limit operator is unbounded with domain in $H^{\alpha/2}(\mathbb{R}^d)$, contrasting with the bounded prelimit operators on $L^2(\mathbb{R}^d)$. The results extend to kernels comparable to $L(|z|)/|z|^{d+\alpha}$ with slowly varying $L$, and align with prior work on stable-like homogenization while handling integrable, heavy-tailed kernels. The approach yields strong resolvent convergence in $L^2$, and explicit form of the effective operator via the mean of $\Lambda$ and the angular density $k$, providing a rigorous description of the homogenized limit for a broad class of nonlocal jump processes. The findings have implications for modeling anomalous diffusion in periodic media where jumps follow stable-like laws. (All mathematical expressions are stated with explicit $\;$LaTeX-style notation.)
Abstract
The paper deals with periodic homogenization of nonlocal symmetric convolution type operators in $L^2(\mathbb R^d)$, whose kernel is the product of a density that belongs to the domain of attraction of an $α$-stable law and a rapidly oscillating positive periodic function. Assuming that the local oscillation of the said density satisfies a proper upper bound at infinity, we prove homogenization result for the studied family of operators.