Before revision:  For the case where the stochastic solution may not be smooth, we characterize it as a limit of smooth  stochastic solutions associated with some approximating Cauchy problems .
After revision:  For the case where the stochastic solution may not be smooth, we characterize it as a limit of smooth stochastic solutions . In deriving the main results, we discover a new, probabilistic proof of Kotani's criterion for martingality of a one-dimensional diffusion in natural scale .