By observing that the fractional Caputo derivative of order α ∈ (0, 1) can be expressed in terms of a multiplicative convolution operator, we introduce and study a class of such operators which also have the same self-similarity property as the Caputo derivative. We proceed by identifying a subclass which is in bijection with the set of Bernstein functions and we provide several representations of their eigenfunctions, expressed in terms of the corresponding Bernstein function, that generalize the Mittag-Leffler function. Each eigenfunction turns out to be the Laplace transform of the right-inverse of a non-decreasing self-similar Markov process associated via the so-called Lamperti mapping to this Bernstein function. Resorting to spectral theoretical arguments, we investigate the generalized Cauchy problems, defined with these self-similar multiplicative convolution operators. In particular, we provide both a stochastic representation, expressed in terms of these inverse processes, and an explicit representation, given in terms of the generalized Mittag-Leffler functions, of the solution of these self-similar Cauchy problems. This work could be seen as an-in depth analysis of a specific class, the one with the self-similarity property, of the general inverse of increasing Markov processes introduced in [15].

中文翻译：

---

自相似柯西问题和广义Mittag-Leffler函数

通过观察阶α∈（0，1）的分数Caputo导数可以用乘积卷积算子表示，我们引入并研究了一类这类算子，它们也具有与Caputo导数相同的自相似性。我们通过确定与伯恩斯坦函数集合双射的子类来进行，并提供了它们的本征函数的几种表示形式，以相应的伯恩斯坦函数表示，概括了Mittag-Leffler函数。每个本征函数都变成了一个非递减的自相似马尔可夫过程的右逆的拉普拉斯变换，该过程通过所谓的Lamperti映射与该伯恩斯坦函数相关联。借助频谱理论论证，我们研究了广义柯西问题，用这些自相似乘法卷积运算符定义。尤其是，我们既提供了用这些逆过程表示的随机表示，又提供了用广义Mittag-Leffler函数给出的这些自相似柯西问题的解决方案的显式表示。这项工作可以看作是对特定类的一种深入分析，该类具有自相似性，是对文献[15]中提出的增加马尔可夫过程的逆的一般反演。