Symmetric Stable Processes on Amenable Groups
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866914777974964224 |
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| author | Avraham-Re'em, Nachi |
| author_facet | Avraham-Re'em, Nachi |
| contents | We show that if $G$ is a countable amenable group, then every stationary non-Gaussian symmetric $α$-stable (S$α$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 $α$ in (0,2), stationary S$α$S processes that are weakly-mixing but not strongly-mixing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_04159 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Symmetric Stable Processes on Amenable Groups Avraham-Re'em, Nachi Probability Dynamical Systems Group Theory 60G52, 60G10, 37A40, 37A50, 43A07 We show that if $G$ is a countable amenable group, then every stationary non-Gaussian symmetric $α$-stable (S$α$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 $α$ in (0,2), stationary S$α$S processes that are weakly-mixing but not strongly-mixing. |
| title | Symmetric Stable Processes on Amenable Groups |
| topic | Probability Dynamical Systems Group Theory 60G52, 60G10, 37A40, 37A50, 43A07 |
| url | https://arxiv.org/abs/2205.04159 |