We discuss the extension of solutions beyond a finite blowup time, i.e., the time at which the system ceases to be Lipschitz continuous. For larger times solutions are defined first by using a (physically motivated) regularization of equations and then taking the limit of a vanishing regularization parameter. We report on several generic situations when such a limit leads to stochastic solutions defining uniquely a probability to choose one or the other (non-unique) path. Moreover, such solutions appear to be independent of the details of regularization procedure, thus, following from the properties of original ideal system alone. In this talk, we provide some rigorous results for systems of ODEs with singularities by using the methods of dynamical system theory applied to renormalized equations. Then, we demonstrate numerical results confirming that such scenarios are realized in infinite dimensional models of hydrodynamic turbulence. Part of the results is a joint work with T.D.Drivas.