  We study some functionals associated with a process driven by a fractional
boundary value problem (FBVP for short). By FBVP we mean a Cauchy problem with
boundary condition written in terms of a fractional equation, that is an
equation involving time-fractional derivative in the sense of Caputo. We focus
on lifetimes and additive functionals characterizing the boundary conditions.
We show that the corresponding additive functionals are related to the
fractional telegraph equations. Moreover, the fractional order of the
derivative gives a unified condition including the elastic and the sticky cases
among the others.
