# Censored Stable Subordinators and Fractional Derivatives

## TL;DR

This paper introduces a new censored fractional derivative based on the Caputo derivative, linking it to a censored stable process, and explores its properties, solutions, and implications for sub-diffusion models.

## Contribution

It defines a novel censored fractional derivative, establishes its connection to a censored stable process, and analyzes its properties and applications in fractional relaxation and sub-diffusion.

## Key findings

- The censored fractional derivative is the Feller generator of the censored stable process.
- The inverse of the censored fractional derivative has a series representation used for initial value problems.
- The relaxation equation's solution exhibits algebraic decay faster than the Caputo case, indicating a new fractional relaxation regime.

## 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.