Stochastic flows for H\"older drifts and transport/continuity equations with noise

TL;DR

This paper establishes the existence of stochastic flows for SDEs with Hölder continuous drifts using a Zvonkin transformation, and applies this to prove well-posedness of stochastic transport and continuity equations with irregular velocity fields.

Contribution

It introduces a method to construct stochastic flows for SDEs with low regularity drifts and applies this to analyze stochastic PDEs with transport noise.

Findings

Existence of stochastic flows for SDEs with Hölder drifts.

Well-posedness of stochastic transport equations with irregular velocity fields.

Well-posedness of stochastic continuity equations with transport noise.

Abstract

We prove existence of a stochastic flow of diffeomorphisms generated by SDEs with drift in $L^q_t C^{0, \alpha}_x$ 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 $BV_\text{loc}$-solutions of stochastic transport equations and weak solutions of stochastic continuity equations with so-called transport noise and velocity fields in $L^q_t C^{0, \alpha}_x$. For both equations, solutions may fail to be unique in the deterministic setting.