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 Hoelder 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 several examples of random walks on nilpotent covering graphs with explicit computations.
Highlights
There are many interests in the study of random walks on infinite graphs in many branches of mathematics such as probability theory, harmonic analysis, geometry, graph theory and group theory
By imposing an additional natural condition (A3), we prove a functional central limit theorems (CLTs) in a Hölder space over http://www.imstat.org/ejp/
We show the tightness of the family of probability measures induced by the G-valued stochastic processes given by the geodesic interpolation of the given random walk (Lemma 4.5)
Summary
There are many interests in the study of random walks on infinite graphs in many branches of mathematics such as probability theory, harmonic analysis, geometry, graph theory and group theory. We refer to Alexopoulos [1, 2], Breuillard [7], Diaconis–Hough [13] and Hough [22] for local CLTs on nilpotent Lie groups In view of these developments, we study the long time behavior of random walks on a covering graph X whose covering transformation group Γ is a finitely generated group of polynomial volume growth. If the dependence of C and O(·) are significant, we denote them like C(N ) and ON (·), respectively
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