Quenched invariance principle for random walks in random environments admitting a cycle decomposition
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915013447385088 |
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| author | Deuschel, Jean-Dominique Slowik, Martin Weng, Weile |
| author_facet | Deuschel, Jean-Dominique Slowik, Martin Weng, Weile |
| contents | We study a class of non-reversible, continuous-time random walks in random environments on $\mathbb{Z}^d$ that admit a cycle representation with finite cycle length. The law of the transition rates, taking values in $[0, \infty)$, is assumed to be stationary and ergodic with respect to space shifts. Moreover, the transition rate from $x$ to $y$, denoted by $c^ω(x,y)$, is a superposition of non-negative random weights on oriented cycles that contain the edge $(x,y)$. We prove a quenched invariance principle under moment conditions that are comparable to the well-known p-q moment condition of Andres, Deuschel, and Slowik [2] for the random conductance model. A key ingredient in proving the sublinearity is an energy estimate for the non-symmetric generator. Our result extends that of Deuschel and Kösters [12] beyond strong ellipticity and bounded cycle lengths. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06861 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quenched invariance principle for random walks in random environments admitting a cycle decomposition Deuschel, Jean-Dominique Slowik, Martin Weng, Weile Probability Analysis of PDEs We study a class of non-reversible, continuous-time random walks in random environments on $\mathbb{Z}^d$ that admit a cycle representation with finite cycle length. The law of the transition rates, taking values in $[0, \infty)$, is assumed to be stationary and ergodic with respect to space shifts. Moreover, the transition rate from $x$ to $y$, denoted by $c^ω(x,y)$, is a superposition of non-negative random weights on oriented cycles that contain the edge $(x,y)$. We prove a quenched invariance principle under moment conditions that are comparable to the well-known p-q moment condition of Andres, Deuschel, and Slowik [2] for the random conductance model. A key ingredient in proving the sublinearity is an energy estimate for the non-symmetric generator. Our result extends that of Deuschel and Kösters [12] beyond strong ellipticity and bounded cycle lengths. |
| title | Quenched invariance principle for random walks in random environments admitting a cycle decomposition |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2411.06861 |