Let $Z = (Z_t)_{t \geq 0}$ be the Rosenblatt process with Hurst index $H \in (1/2, 1)$. We prove joint continuity for the local time of $Z$, and establish Holder conditions for the local time. These results are then used to study the irregularity of the sample paths of $Z$. Based on analogy with similar known results in the case of fractional Brownian motion, we believe our results are sharp. A main ingredient of our proof is a rather delicate spectral analysis of arbitrary linear combinations of integral operators, which arise from the representation of the Rosenblatt process as an element in the second chaos.

中文翻译：

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Rosenblatt 过程的本地时间和样本路径属性

令 $Z = (Z_t)_{t \geq 0}$ 是具有 Hurst 指数 $H \in (1/2, 1)$ 的 Rosenblatt 过程。我们证明了 $Z$ 当地时间的联合连续性，并建立了当地时间的 Holder 条件。然后将这些结果用于研究 $Z$ 样本路径的不规则性。基于在分数布朗运动的情况下与类似已知结果的类比，我们相信我们的结果是尖锐的。我们证明的一个主要成分是对积分算子的任意线性组合进行相当精细的谱分析，这些组合源于将 Rosenblatt 过程表示为第二混沌中的一个元素。