In this work, we provide sharp bounds on the rate of growth of maximal moments for stationary symmetric stable random fields when the underlying nonsingular group action (or its restriction to a suitable lower rank subgroup) has a nontrivial dissipative component. We also investigate the relationship between this rate of growth and the path regularity properties of self-similar stable random fields with stationary increments, and establish uniform modulus of continuity of such fields. In the process, a new notion of weak effective dimension is introduced for stable random fields and is connected to maximal moments and path properties.

中文翻译：

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稳定随机场的最大矩和一致的连续模量

在这项工作中，当潜在的非奇异群作用（或其对适当的较低阶次群的限制）具有非平凡的耗散分量时，我们为平稳对称稳定随机场的最大矩的增长率提供了清晰的界限。我们还研究了此增长率与具有相似增量的自相似稳定随机场的路径规律性之间的关系，并建立了此类场的均匀连续模量。在此过程中，为稳定的随机场引入了有效维数弱的新概念，并将其与最大力矩和路径特性相关联。