删失稳定从属算子与分数阶导数
Censored Stable Subordinators and Fractional Derivatives
摘要：
基于$(0,1)$中流行的$β$阶Caputo分数阶导数，定义了正半直线$\mathbb R_+$上的截尾分数阶导数.证明了该导数是$\mathbb R_+$中截尾（或复活）$β$ -稳定过程的Feller生成元.我们提供了一个系列表示的逆这个截尾分数阶导数，我们用它来研究一般的截尾初值问题。然后，我们能够证明，这个截尾过程击中边界在有限的时间$τ_\infty$，其期望是成比例的第一次通过时间的$β$稳定从属。我们还表明，删失松弛方程解的拉普拉斯变换的$τ_\infty$。这个松弛解被证明是一个完全单调的系列，代数衰减一阶比其卡普托对应，导致，令人惊讶的是，一个新的制度的分数松弛模型。最后，我们讨论了这项工作如何识别一个新的子扩散模型。
英文摘要：
Based on the popular Caputo fractional derivative of order $β$ in $(0,1)$, we define the censored fractional derivative on the positive half-line $\mathbb R_+$. This derivative proves to be the Feller generator of the censored (or resurrected) decreasing $β$-stable process in $\mathbb R_+$. We provide a series representation for the inverse of this censored fractional derivative, which we use to study general censored initial value problems. We are then able to prove that this censored process hits the boundary in a finite time $τ_\infty$, whose expectation is proportional to that of the first passage time of the $β$-stable subordinator. We also show that the censored relaxation equation is solved by the Laplace transform of $τ_\infty$. This relaxation solution proves to be a completely monotone series, with algebraic decay one order faster than its Caputo counterpart, leading, surprisingly, to a new regime of fractional relaxation models. Lastly, we discuss how this work identifies a new sub-diffusion model.
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