Georgia Institute of Technology College of Sciences

We consider the small noise perturbation (in the Ito sense) of a one dimensional ODE. We study the case in which the ODE has not unique solution, but the SDE does. A particular setting of this sort is studied and the properties of the solution are obtained when the noise level vanishes. We relate this to give an example of a 1-dimensional transport equation without uniqueness of weak solution. We show how by a suitable random noise perturbation, the stochastic equation is well posed and study what the limit is when the noise level tends to zero.