Stationary symmetric $\alpha$-stable ($S \alpha S$) processes is an important class of stochastic processes, including Gaussian processes, Cauchy processes and Lévy processes. In an analogy to that the ergodicity of a Gaussian process is determined by its spectral measure, it was shown by Rosinski and Samorodnitsky that the ergodicity of a stationary $S \alpha S$ process is characterized by its spectral representation. While this result is known when the process is indexed by $\mathbb{Z}$ or $\mathbb{R}$, the classical techniques fail when it comes to processes indexed by non-Abelian groups and it was an open question whether the ergodicity of stationary $S \alpha S$ processes indexed by amenable groups admits a similar characterization.

In this talk I will introduce the fundamentals of stable processes, the ergodic theory behind their spectral representation, and the key ideas of the characterization of the ergodicity for processes indexed by amenable groups. If time permits, I will explain how to use a recent construction of A. Danilenko in order to prove the existence of weakly-mixing but not strongly-mixing stable processes indexed by many groups (Abelian groups, Heisenberg group).