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Censored stable subordinators and fractional derivatives

  • Qiang Du , Lorenzo Toniazzi EMAIL logo und Zirui Xu
Veröffentlicht/Copyright: 23. August 2021
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Abstract

Based on the popular Caputo fractional derivative of order β in (0, 1), we define the censored fractional derivative on the positive half-line ℝ+. This derivative proves to be the Feller generator of the censored (or resurrected) decreasing β-stable process in ℝ+. We provide a series representation for the inverse of this censored fractional derivative. We are then able to prove that this censored process hits the boundary in a finite time τ, 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. 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.

Acknowledgements

The authors thank their institutions for the support, under Grant NSF DMS-2012562, the ARO MURI Grant W911NF-15-1-0562 and the Marsden Fund administered by the Royal Society of New Zealand. The authors also thank Professors Kai Diethelm, Thomas Simon, Zhen-Qing Chen, Krzysztof Bogdan and Zhi Zhou for many helpful discussions.

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Received: 2020-12-31
Published Online: 2021-08-23
Published in Print: 2021-08-26

© 2021 Diogenes Co., Sofia

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