Remarks on the stochastic transport equation with Hölder drift

We consider a stochastic linear transport equation with a globally Hölder 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 Flandoli-Gubinelli-Priola (2009) about well-posedness. Moreover, we prove a stability property for the solution with respect to the initial datum.

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