Quantitative stochastic homogenization for long-range random walks with critical jump index
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908988051816448 |
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| author | Chen, Xin Gu, Chenlin Wang, Jian |
| author_facet | Chen, Xin Gu, Chenlin Wang, Jian |
| contents | 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 $α$-th moment for all $α\in (0,2)$, 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 $α$-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 $(\log k)^{-\frac{1}{2}+\frac{1}{2(d-2)}+\varepsilon}$ with any $\varepsilon>0$ for all $d>3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21162 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantitative stochastic homogenization for long-range random walks with critical jump index Chen, Xin Gu, Chenlin Wang, Jian Probability 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 $α$-th moment for all $α\in (0,2)$, 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 $α$-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 $(\log k)^{-\frac{1}{2}+\frac{1}{2(d-2)}+\varepsilon}$ with any $\varepsilon>0$ for all $d>3$. |
| title | Quantitative stochastic homogenization for long-range random walks with critical jump index |
| topic | Probability |
| url | https://arxiv.org/abs/2604.21162 |