  The Tolman-Oppenheimer-Snyder description gives exact analytical solutions
for an Einstein-matter system describing total gravitational collapse of a
zero-pressure perfect-fluid sphere, representing a massive star which has
exhausted its nuclear fuel. The star collapses to a point of infinite density
within a finite comoving proper time interval $[0,t_{*}]$, and the exterior
metric matches the Schwarzchild black hole metric. The description is
re-expressed in terms of a 'density function'
$u(t)=(\rho(t)/\rho_{o}))^{1/3}=R^{-1}(t)$ for initial density
$u_{0}=R^{-1}(0)=1$ and radius $R(0)$, whereby the general-relativistic
formulation reduces to an autonomous nonlinear ODE for $u(t)$. The solution
blows up or is singular at $t=t_{*}=\pi/2(8\pi G/3\rho_{o})^{1/2}$. The blowup
interval $[0,t_{*}]$ is partitioned into domains
$[0,t_{\epsilon}]\bigcup[t_{\epsilon},t_{*}]$,with
$t_{*}=t_{\epsilon}+|\epsilon|$ and $|\epsilon|\ll 1$, so that $t_{\epsilon}$
can be infinitesimally close to $t_{*}$. Randomness or 'stochastic control' is
introduced via the 'switching on' of specific (white-noise) perturbations at
$t=t_{\epsilon}$. Hybrid nonlinear ODES-SDES are then 'engineered' over the
partition. Within the Ito interpretation, the resulting density function
diffusion $\overline{u(t)}$ is proved to be a martingale whose supremum,
volatility and higher-order moments are finite, bounded and singularity free
for all finite $t>t_{\epsilon}$. The collapse is (comovingly) eternal but never
becomes singular. Extensive and rigorous boundedness and no-blowup criteria are
established via various methods, and blowup probability is always zero. The
density singularity is therefore smoothed or 'noise-suppressed'. Within the
Stratanovitch interpretation, the singularity formation probability is unity;
however, null recurrence ensures the expected comoving time for this to occur
is now infinite.
