  We study the connection between PDEs and L\'{e}vy processes running with
clocks given by time-changed Poisson processes with stochastic drifts. The
random times we deal with are therefore given by time-changed Poissonian jumps
related to some Frobenious-Perron operators $K$ associated to random
translations. Moreover, we also consider their hitting times as a random clock.
Thus, we study processes driven by equations involving time-fractional
operators (modelling memory) and fractional powers of the difference operator
$I-K$ (modelling jumps). For this large class of processes we also provide, in
some cases, the explicit representation of the transition probability laws. To
this aim, we show that a special role is played by the translation operator
associated to the representation of the Poisson semigroup.
