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Symmetric stable processes on amenable groups

Volume 271 / 2023

Nachi Avraham-Re’em 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


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.


  • Nachi Avraham-Re’emEinstein Institute of Mathematics
    The Hebrew University of Jerusalem
    Givat Ram, 9190401 Jerusalem, Israel

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