In this paper, we perform the joint study of scale-invariant operators and self-similar processes of complex order. More precisely, we introduce general families of scale-invariant complex-order fractional-derivation and integration operators by constructing them in the Fourier domain. We analyze these operators in detail, with special emphasis on the decay properties of their output. We further use them to introduce a family of complex-valued stable processes that are self-similar with complex-valued Hurst exponents. These random processes are expressed via their characteristic functionals over the Schwartz space of functions. They are therefore defined as generalized random processes in the sense of Gel'fand. Beside their self-similarity and stationarity, we study the Sobolev regularity of the proposed random processes. Our work illustrates the strong connection between scale-invariant operators and self-similar processes, with the construction of adequate complex-order scale-invariant integration operators being preparatory to the construction of the random processes.

中文翻译：

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复阶尺度不变算子和自相似过程


在本文中，我们对尺度不变算子和复阶自相似过程进行了联合研究。更准确地说，我们通过在傅里叶域中构造尺度不变的复阶分数导数和积分算子来引入它们的一般族。我们详细分析这些运算符，特别强调其输出的衰减特性。我们进一步使用它们来引入一系列与复值赫斯特指数自相似的复值稳定过程。这些随机过程通过其在施瓦茨函数空间上的特征泛函来表达。因此它们被定义为盖尔凡德意义上的广义随机过程。除了它们的自相似性和平稳性之外，我们还研究了所提出的随机过程的 Sobolev 规律。我们的工作说明了尺度不变算子和自相似过程之间的紧密联系，构造了足够的复阶尺度不变积分算子，为随机过程的构造做好了准备。