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.

在本文中，作为我们之前论文（Ishiwata等人2018）的延续，我们从离散几何分析的角度研究了另一种关于幂等覆盖图上非对称随机游动的中心极限定理（CLT）由Kotani和Sunada撰写。我们介绍了一个随机游动的单参数族，该族插值在原始非对称随机游动和对称对称游动之间。我们首先通过离散谐波图将幂等覆盖图实现为幂等李群，从而证明了随机游动族的半群CLT。极限扩散半群是由均质的次拉普拉斯算子生成的，该渐近子在无幂李群上具有渐近方向的恒定漂移，该群装备了与对称随机游走相关的阿尔巴涅斯度量。接下来，我们通过结合半群CLT，标准mar技术和受粗糙路径理论启发的新颖的路径论证，证明了幂等李群上Hölder空间中的CLT（即Donsker型不变性原理）。应用校正器方法，我们最终将这些CLT扩展到实现不一定是谐波的情况。