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2020 Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part I
Satoshi Ishiwata, Hiroshi Kawabi, Ryuya Namba
Electron. J. Probab. 25: 1-46 (2020). DOI: 10.1214/20-EJP486

Abstract

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.

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Satoshi Ishiwata. Hiroshi Kawabi. Ryuya Namba. "Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part I." Electron. J. Probab. 25 1 - 46, 2020. https://doi.org/10.1214/20-EJP486

Information

Received: 25 October 2019; Accepted: 23 June 2020; Published: 2020
First available in Project Euclid: 22 July 2020

zbMATH: 07252718
MathSciNet: MR4125791
Digital Object Identifier: 10.1214/20-EJP486

Subjects:
Primary: 22E25, 60F17, 60G50, 60J10

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