Censored Stable Subordinators and Fractional Derivatives

Abstract

Based on the popular Caputo fractional derivative of order $\beta$ 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 $\beta$-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 ${\tau }_{\infty }$, whose expectation is proportional to that of the first passage time of the $\beta$-stable subordinator. We also show that the censored relaxation equation is solved by the Laplace transform of ${\tau }_{\infty }$. 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.

arXiv comment47 pages; minor typos corrected; publisher's note added

References in corpus (10)

• Ten equivalent definitions of the fractional Laplace operator
• Space-time fractional diffusion on bounded domains
• What Is the Fractional Laplacian?
• Hitting distributions of alpha-stable processes via path censoring and self-similarity
• On fully mixed and multidimensional extensions of the Caputo and Riemann-Liouville derivatives, related Markov processes and fractional differential equations
• Generalised Fractional Evolution Equations of Caputo Type
• Predicting the ultimate supremum of a stable Lévy process with no negative jumps
• On the probabilistic approach to the solution of generalized fractional differential equations of Caputo and Riemann-Liouville type
• Censored Stable Subordinators and Fractional Derivatives
• Stochastic Schemes of Dielectric Relaxation in Correlated-Cluster Systems

Cited by in corpus (1)

• Censored Stable Subordinators and Fractional Derivatives