Random walks on rank one symmetric spaces of noncompact type
Fedor Gnetov, Valentin Konakov
TL;DR
The paper extends classical limit theorems to random walks on rank-one noncompact symmetric spaces by introducing a non-Euclidean summation framework that generalizes Möbius addition. It leverages harmonic analysis of spherical functions to prove a law of large numbers, a central limit theorem with the heat kernel of the Laplace-Beltrami operator as the limiting law, and a local limit theorem with explicit convergence rates. The results identify the heat kernel as the natural non-Euclidean analogue of the normal distribution and provide precise variance scaling via the radial component of the walk. This work links geometry, representation theory, and probability to advance understanding of stochastic processes on curved spaces.
Abstract
We establish a central limit theorem, a local limit theorem, and a law of large numbers for a natural random walk on a symmetric space $M$ of non-compact type and rank one. This class of spaces, which includes the complex and quaternionic hyperbolic spaces and the Cayley hyperbolic plane, generalizes the real hyperbolic space $\mathbb{H}^{n}$. Our approach introduces a unified algebraic framework that generalizes the Möbius addition, previously used for the constant curvature case, to define the random walk via a non-Euclidean summation of variables. We demonstrate that the renormalized walk converges to the heat kernel associated with the Laplace-Beltrami operator on $M$, which plays the role of the limiting normal law. The proofs leverage the harmonic analysis of spherical functions on symmetric spaces. To the best of our knowledge, these results are new in the context of rank one symmetric spaces.