Malliavin and flow regularity of SDEs. Application to the study of   densities and the stochastic transport equation
  In this work we present a condition for the regularity, in both space and Malliavin sense, of strong solutions to SDEs driven by Brownian motion. We conjecture that this condition is optimal. As a consequence, we are able to improve the regularity of densities of such solutions.   We also apply these results to construct a classical solution to the stochastic transport equation when the drift is Lipschitz. 
David Banos and Torstein Nilssen
math.PR