Remarks on the stochastic transport equation with Holder drift

We consider a stochastic linear transport equation with a globally H\"older continuous and bounded vector field. Opposite to what happens in the deterministic case where shocks may appear, we show that the unique solution
starting with a $C^1$-initial condition remains of class $C^1$ in space. We also improve some results of \citeFGP about well-posedness. Moreover, we prove a stability property for the solution with respect to the initial datum.