Before revision:  The analysis is restricted to the one-dimensional case, and we only require a fairly mild condition on the diffusion coefficient, namely to be non-zero in a point of discontinuity  , while we need to impose certain structural assumptions on the measure-dependence of the drift.
After revision:  The analysis is restricted to the one-dimensional case, and we only require a fairly mild condition on the diffusion coefficient, namely to be non-zero in a point of discontinuity of the drift , while we need to impose certain structural assumptions on the measure-dependence of the drift.