Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness

Linear stochastic transport and continuity equations with drift in critical Lp spaces are considered. A result of Sobolev regularity of solutions is proved, false for the corresponding deterministic equations. Thus noise prevents shocks for transport equation and singularities in the density for continuity equation, starting from smooth initial conditions. The technique needed to reach the critical case is new and based on parabolic equations satisfied by moments of first derivatives of the solution, opposite to previous works based on stochastic flows. The approach extends to higher order derivatives under more regularity of the drift term. By a duality approach, the results are then applied to prove uniqueness of weak solutions to linear stochastic continuity and transport equations and certain well posedness results for the associated stochastic differential equation (sDE) (roughly speaking, existence and uniqueness of flows and their Cα regularity, strong uniqueness for the sDE when the initial datum has diffuse law). Finally, we show two types of examples: on the one hand, we present well-posed sDEs, when the corresponding ODEs are ill-posed, and on the other hand, we give a counterexample in the supercritical case.