  We consider the dynamics of a gas of free bosons within a semi-classical
Fokker-Planck equation for which we give a physical justification. In this
context, we find a striking similarity between the Bose-Einstein condensation
in the canonical ensemble, and the gravitational collapse of a gas of classical
self-gravitating Brownian particles. The paper is mainly devoted to the
complete study of the Bose-Einstein ``collapse'' within this model. We find
that at the Bose-Einstein condensation temperature $T_c$, the chemical
potential $\mu(t)$ vanishes exponentially with a universal rate that we compute
exactly. Below $T_c$, we show analytically that $\sqrt{\mu(t)}$ vanishes
linearly in a finite time $t_{coll}$. After $t_{coll}$, the mass of the
condensate grows linearly with time and saturates exponentially to its
equilibrium value for large time. We also give analytical results for the
density scaling functions, for the corrections to scaling, and for the
exponential relaxation time. Finally, we find that the equilibration time
(above $T_c$) and the collapse time $t_{coll}$ (below $T_c$), both behave like
$-T_c^{-3}\ln|T-T_c|$, near $T_c$.
