Censored fractional Bernstein derivatives and stochastic processes

In this paper, we define the censored fractional Bernstein derivative on the positive half line $(0,\mathrm{\infty })$ based on the Bernstein Riemann–Liouville fractional derivative. This derivative can be shown to be the generator of the censored subordinator by solving a resolvent equation. We also show that the censored subordinator hits the boundary in finite time under certain conditions.