Symmetric stable processes on amenable groups
Volume 271 / 2023
Studia Mathematica 271 (2023), 187-224
MSC: Primary 60G52; Secondary 60G10, 37A40, 37A50, 43A07.
DOI: 10.4064/sm220924-19-2
Published online: 29 March 2023
Abstract
We show that if $G$ is a countable amenable group, then every stationary non-Gaussian symmetric $\alpha $-stable ($S\alpha S$) process indexed by $G$ is ergodic if and only if it is weakly mixing, and it is ergodic if and only if its Rosiński minimal spectral representation is null. This extends previous results for $\mathbb {Z}^d$, and answers a question of P. Roy on discrete nilpotent groups in the range of all countable amenable groups. As a result, we construct on the Heisenberg group and on many Abelian groups, for all $\alpha \in (0,2)$, stationary $S\alpha S$ processes that are weakly mixing but not strongly mixing.
