Symmetric stable processes on amenable groups
Tom 271 / 2023
Streszczenie
We show that if 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.