标题： Hadamard分数布朗运动：路径性质和维纳积分 显示英文标题

标题： Hadamard fractional Brownian motion: path properties and Wiener integration

摘要： 所谓的Hadamard分数阶布朗运动，如Beghin等人（2025）通过Hadamard分数阶算子定义的，是一个高斯过程，它与标准布朗运动（如一维分布）有一些相似的性质。然而，它在许多其他特征上也类似于分数阶布朗运动，例如自相似性、长/短记忆特性、维纳积分表示。Hadamard分数阶布朗运动中的对数核代表了该过程的一个非常具体且有趣的特点。我们在此分析该过程轨迹的一些性质（即霍尔德连续性、准螺旋行为、幂次变分、局部非确定性），这些性质本身就很有趣，并且是针对该过程进行维纳积分的基础。相应的积分已经得到了很好的发展，逆表示也被构建出来。我们将推导出的“乘法Sonine对”应用于Hadamard分数阶布朗运动的再生核希尔伯特空间的处理，并由此建立了一个迭代对数定律。 显示英文摘要

摘要： The so-called Hadamard fractional Brownian motion, as defined in Beghin et al. (2025) by means of Hadamard fractional operators, is a Gaussian process which shares some properties with standard Brownian motion (such as the one-dimensional distribution). However, it also resembles the fractional Brownian motion in many other features as, for instance, self-similarity, long/short memory property, Wiener-integral representation. The logarithmic kernel in the Hadamard fractional Brownian motion represents a very specific and interesting aspect of this process. Our aim here is to analyze some properties of the process' trajectories (i.e. H\"{o}lder continuity, quasi-helix behavior, power variation, local nondeterminism) that are both interesting on their own and serve as a basis for the Wiener integration with respect to it. The respective integration is quite well developed, and the inverse representation is also constructed. We apply the derived ``multiplicative Sonine pairs'' to the treatment of the Reproducing Kernel Hilbert Space of the Hadamard fractional Brownian motion, and, as a result, we establish a law of iterated logarithm.