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.