  We consider SDEs with (distributional) drift in negative Besov spaces and
random initial condition and investigate them from two different viewpoints. In
the first part we set up a martingale problem and show its well-posedness.We
then prove further properties of the martingale problem, like continuity with
respect to the drift and the link with the Fokker-Planck equation. We also show
that the solutions are weak Dirichlet processes for which we evaluate the
quadratic variation of the martingale component.In the second part we identify
the dynamics of the solution of the martingale problemby describing the proper
associated SDE.Under suitable assumptions we show equivalence with the solution
to the martingale problem.
