  We complete the physical picture for the evolution of a high-energy jet
propagating through a weakly-coupled quark-gluon plasma by investigating the
thermalization of the soft components of the jet. We argue that the following
scenario should hold: the leading particle emits a significant number of
mini-jets which promptly evolve via quasi-democratic branchings and thus
degrade into a myriad of soft gluons, with energies of the order of the medium
temperature T. Via elastic collisions with the medium constituents, these soft
gluons relax to local thermal equilibrium with the plasma over a time scale
which is considerably shorter than the typical lifetime of the mini-jet. The
thermalized gluons form a tail which lags behind the hard components of the
jet. We support this scenario, first, via parametric arguments and, next, by
studying a simplified kinetic equation, which describes the jet dynamics in
longitudinal phase-space. We solve the kinetic equation using both
(semi-)analytical and numerical methods. In particular, we obtain the first
exact, analytic, solutions to the ultrarelativistic Fokker-Planck equation in
one-dimensional phase-space. Our results confirm the physical picture
aforementioned and demonstrate the quenching of the jet via multiple branching
followed by the thermalization of the soft gluons in the cascades.
