  We study Lagrangian particles, with Stokes numbers, $\textrm{St}=0.5$, $3$,
and $6$, transported by homogeneous and isotropic turbulent flows. Based on
direct numerical simulations with point-like inertial particles, we identify
stochastic equations describing the multi-scale cascade process. We show that
the Markov property is valid for a finite step size larger than a
$\textrm{St}$-dependent Einstein-Markov memory length. The formalism allows
estimation of the entropy of the particles' Lagrangian trajectories. Integral,
as well as detailed fluctuation theorems are fulfilled. Entropy consuming
trajectories are related to specific local accelerations of the particles and
may be seen as reverse cascade processes.
