TY - JOUR

T1 - Central Limit Theorems for Non-Symmetric Random Walks on Nilpotent Covering Graphs

T2 - Part II

AU - Ishiwata, Satoshi

AU - Kawabi, Hiroshi

AU - Namba, Ryuya

N1 - Funding Information:
The authors are grateful to Professor Shoichi Fujimori for making pictures of the 3-dimensional Heisenberg hexagonal lattice and kindly allowing them to use these pictures in the present paper. They would like to thank Professor Seiichiro Kusuoka for reading of our manuscript carefully and for giving valuable comments. They also thank the anonymous referee who significantly helped to improve the quality of the present paper. A part of this work was done during the stay of R. N. at Hausdorff Center for Mathematics, Universität Bonn in March 2017 with the support of research fund of Research Institute for Interdisciplinary Science, Okayama University. He would like to thank Professor Massimiliano Gubinelli for warm hospitality and helpful discussions. S. I. was partially supported by JSPS Grant-in-Aid for Young Scientists (B) No. 25800034 and JSPS Grant-in-Aid for Scientific Research (C) No. 17K05215. H.K. was supported by JSPS Grant-in-Aid for Scientific Research (C) No. 26400134 and (C) No. 17K05300. R. N. was partially supported by JSPS Research Fellowships for Young Scientists No. 18J10225. The present paper is based on a part of R.N.’s Ph.D. thesis [ 24 ].

PY - 2020

Y1 - 2020

N2 - In the present paper, as a continuation of our preceding paper (Ishiwata et al. 2018), we study another kind of central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a view point 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 Hölder 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.

AB - In the present paper, as a continuation of our preceding paper (Ishiwata et al. 2018), we study another kind of central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a view point 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 Hölder 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.

KW - Central limit theorem

KW - Discrete geometric analysis

KW - Nilpotent covering graph

KW - Non-symmetric random walk

KW - Rough path theory

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U2 - 10.1007/s11118-020-09851-7

DO - 10.1007/s11118-020-09851-7

M3 - Article

AN - SCOPUS:85089453702

JO - Potential Analysis

JF - Potential Analysis

SN - 0926-2601

ER -