Stochastic flows for Hölder drifts and transport/continuity equations with noise

Abstract

We prove existence of a stochastic flow of diffeomorphisms generated by SDEs with drift in $L_t^q{C}_x^{0,\alpha }$ for any $q\in [2,\infty )$ and $\alpha \in (0,1)$. This result is achieved using a Zvonkin-type transformation for the SDE. As a key intermediate step, well-posedness and optimal regularity for a class of parabolic PDEs related to the transformation is established. Using the existence of a differentiable stochastic flow, we prove well-posedness of $B{V}_{\text{loc}}$-solutions of stochastic transport equations and weak solutions of stochastic continuity equations with so-called transport noise and velocity fields in $L_t^q{C}_x^{0,\alpha }$. For both equations, solutions may fail to be unique in the deterministic setting.

29 pages