Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part II
| Autor: | Ishiwata, Satoshi, Kawabi, Hiroshi, Namba, Ryuya |
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| Rok vydání: | 2018 |
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| Zdroj: | Potential Analysis 55 (2021), 127-166 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s11118-020-09851-7 |
| Popis: | In the present paper, as a continuation of our preceding paper [10], we study another kind of central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a viewpoint of discrete geometric analysis developed by Kotani and Sunada. We introduce a one-parameter family of random walks which interpolates between the original non-symmetric random walk and the symmetrized one. We first prove a semigroup CLT for the family of random walks by realizing the nilpotent covering graph into a nilpotent Lie group via discrete harmonic maps. The limiting diffusion semigroup is generated by the homogenized sub-Laplacian with a constant drift of the asymptotic direction on the nilpotent Lie group, which is equipped with the Albanese metric associated with the symmetrized random walk. We next prove a functional CLT (i.e., Donsker-type invariance principle) in a Holder space over the nilpotent Lie group by combining the semigroup CLT, standard martingale techniques, and a novel pathwise argument inspired by rough path theory. Applying the corrector method, we finally extend these CLTs to the case where the realizations are not necessarily harmonic. Comment: 41 pages, 2 figures. arXiv admin note: text overlap with arXiv:1806.03804" |
| Databáze: | arXiv |
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