论文标题
小组测量空间构建，千古和$ w^\ ast $ - rigities稳定随机字段
Group measure space construction, ergodicity and $W^\ast$-rigidity for stable random fields
论文作者
论文摘要
这项工作发现了概率理论（稳定随机场）和冯·诺伊曼代数之间的新联系。可以确定的是，与最小表示相对应的小组测量空间构建是固定对称$α$稳定（S $α$ s）随机字段的不变，任何可数组$ g $。当$ g = \ mathbb {z}^d $时，我们表征了固定s $α$α$ s田地的Ergodicity（也是绝对的非共性性），就此交叉产品von Neumann代数的中心分解而言，来自任何（不一定是最小）Rosinski的代表。这表明，对于此类的字段，Ergodicity（或完全没有它）是$ w^\ ast $ rigid属性（合适的意义上）。我们所有的结果都具有固定最大稳定随机字段的类似物。
This work discovers a novel link between probability theory (of stable random fields) and von Neumann algebras. It is established that the group measure space construction corresponding to a minimal representation is an invariant of a stationary symmetric $α$-stable (S$α$S) random field indexed by any countable group $G$. When $G=\mathbb{Z}^d$, we characterize ergodicity (and also absolute non-ergodicity) of stationary S$α$S fields in terms of the central decomposition of this crossed product von Neumann algebra coming from any (not necessarily minimal) Rosinski representation. This shows that ergodicity (or the complete absence of it) is a $W^\ast$-rigid property (in a suitable sense) for this class of fields. All our results have analogues for stationary max-stable random fields as well.
