We consider a stochastic linear transport equation with a globally H\"{o}lder 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 \cite{FGP} about well-posedness. Moreover, we prove a stability property for the solution with respect to the initial datum.
Remarks on the stochastic transport equation with Holder drift
PRIOLA, Enrico
2012-01-01
Abstract
We consider a stochastic linear transport equation with a globally H\"{o}lder 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 \cite{FGP} about well-posedness. Moreover, we prove a stability property for the solution with respect to the initial datum.
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