Symmetric Stable Processes on Amenable Groups

TL;DR

This paper extends ergodic and mixing properties of symmetric alpha-stable processes from integer lattices to all countable amenable groups, including the Heisenberg group, revealing new distinctions between weak and strong mixing.

Contribution

It generalizes known results for  lattices to all countable amenable groups and constructs examples of weakly-mixing but not strongly-mixing processes on these groups.

Findings

Ergodicity is equivalent to weak-mixing for non-Gaussian symmetric lpha-stable processes on countable amenable groups.

The spectral representation's nullity characterizes ergodicity in this setting.

Existence of processes that are weakly-mixing but not strongly-mixing on the Heisenberg and Abelian groups.

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 Rosinski minimal spectral representation is null. This extends the results for $\mathbb{Z}^d$, and answers a question of P. Roy on discrete nilpotent groups to the extent 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.