TY - JOUR

T1 - Central limit theorems for non-symmetric random walks on nilpotent covering graphs

T2 - Part i

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 dice lattice and kindly allowing them to use these pictures in the present paper. They would also like to thank Professors Takafumi Amaba, Takahiro Aoyama, Peter Friz, Naotaka Kajino, Atsushi Katsuda, Takashi Kumagai, Seiichiro Kusuoka, Kazumasa Kuwada, Laurent Saloff-Coste and Ryokichi Tanaka for helpful discussions and encouragement. A part of this work was done during the stay of R. N. at Hausdorff Center for Mathematics, Universit?t 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 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 supported by JSPS Research Fellowships for Young Scientists No. 18J10225.

PY - 2020

Y1 - 2020

N2 - In the present paper, we study central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a point of view of discrete geometric analysis developed by Kotani and Sunada. We establish a semigroup CLT for a non-symmetric random walk on a nilpotent covering graph. Realizing the nilpotent covering graph into a nilpotent Lie group through a discrete harmonic map, we give a geometric characterization of the limit semigroup on the nilpotent Lie group. More precisely, we show that the limit semigroup is generated by the sub-Laplacian with a non-trivial drift on the nilpotent Lie group equipped with the Albanese metric. The drift term arises from the non-symmetry of the random walk and it vanishes when the random walk is symmetric. Furthermore, by imposing the “centered condition”, we establish a functional CLT (i.e., Donsker-type invariance principle) in a Hölder space over the nilpotent Lie group. The functional CLT is extended to the case where the realization is not necessarily harmonic. We also obtain an explicit representation of the limiting diffusion process on the nilpotent Lie group and discuss a relation with rough path theory. Finally, we give an example of random walks on nilpotent covering graphs with explicit computations.

AB - In the present paper, we study central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a point of view of discrete geometric analysis developed by Kotani and Sunada. We establish a semigroup CLT for a non-symmetric random walk on a nilpotent covering graph. Realizing the nilpotent covering graph into a nilpotent Lie group through a discrete harmonic map, we give a geometric characterization of the limit semigroup on the nilpotent Lie group. More precisely, we show that the limit semigroup is generated by the sub-Laplacian with a non-trivial drift on the nilpotent Lie group equipped with the Albanese metric. The drift term arises from the non-symmetry of the random walk and it vanishes when the random walk is symmetric. Furthermore, by imposing the “centered condition”, we establish a functional CLT (i.e., Donsker-type invariance principle) in a Hölder space over the nilpotent Lie group. The functional CLT is extended to the case where the realization is not necessarily harmonic. We also obtain an explicit representation of the limiting diffusion process on the nilpotent Lie group and discuss a relation with rough path theory. Finally, we give an example of random walks on nilpotent covering graphs with explicit computations.

KW - Albanese metric

KW - Central limit theorem

KW - Discrete geometric analysis

KW - Modified harmonic realization

KW - Nilpotent covering graph

KW - Non-symmetric random walk

KW - Rough path theory

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U2 - 10.1214/20-EJP486

DO - 10.1214/20-EJP486

M3 - Article

AN - SCOPUS:85089452123

VL - 25

SP - 1

EP - 46

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

M1 - 86

ER -