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The Tolman-Oppenheimer-Snyder description gives exact analytical solutionsfor an Einstein-matter system describing total gravitational collapse of azero-pressure perfect-fluid sphere, representing a massive star which hasexhausted its nuclear fuel. The star collapses to a point of infinite densitywithin a finite comoving proper time interval $[0,t_{*}]$, and the exteriormetric matches the Schwarzchild black hole metric. The description isre-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-relativisticformulation reduces to an autonomous nonlinear ODE for $u(t)$. The solutionblows up or is singular at $t=t_{*}=\pi/2(8\pi G/3\rho_{o})^{1/2}$. The blowupinterval $[0,t_{*}]$ 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' isintroduced via the 'switching on' of specific (white-noise) perturbations at$t=t_{\epsilon}$. Hybrid nonlinear ODES-SDES are then 'engineered' over thepartition. Within the Ito interpretation, the resulting density functiondiffusion $\overline{u(t)}$ is proved to be a martingale whose supremum,volatility and higher-order moments are finite, bounded and singularity freefor all finite $t>t_{\epsilon}$. The collapse is (comovingly) eternal but neverbecomes singular. Extensive and rigorous boundedness and no-blowup criteria areestablished via various methods, and blowup probability is always zero. Thedensity singularity is therefore smoothed or 'noise-suppressed'. Within theStratanovitch interpretation, the singularity formation probability is unity;however, null recurrence ensures the expected comoving time for this to occuris now infinite.

Stochastic Control of Tolman-Oppenheimer-Snyder Collapse of  Zero-Pressure Stars to Black Holes: Rigorous Criteria for Density Bounds and  Singularity Smoothing