Limit theorems for some long range random walks on torsion free nilpotent groups
Zhen-Qing Chen, Takashi Kumagai, Laurent Saloff-Coste, Jian Wang and, Tianyi Zheng

TL;DR
This paper establishes limit theorems for long-range random walks on torsion-free nilpotent groups, showing convergence to stable-like processes on associated Lie groups, with results dependent on the measure used.
Contribution
It introduces a framework for deriving limit theorems for long-range random walks on torsion-free nilpotent groups, including the construction of a limit Lie group and process.
Findings
Convergence of rescaled random walks to stable-like processes on Lie groups.
Construction of a homogeneous nilpotent Lie group associated with the walk.
Establishment of local and functional limit theorems for these walks.
Abstract
We consider a natural class of long range random walks on torsion free nilpotent groups and develop limit theorems for these walks. Given the original discrete group and a random walk driven by a certain type of symmetric probability measure , we construct a homogeneous nilpotent Lie group which carries an adapted dilation structure and a stable-like process which appears in a Donsker-type functional limit theorem as the limit of a rescaled version of the random walk. Both the limit group and the limit process on that group depend on the measure . In addition, the functional limit theorem is complemented by a local limit theorem.
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Taxonomy
TopicsGeometric and Algebraic Topology · Mathematical Dynamics and Fractals · Stochastic processes and statistical mechanics
