Stochastic transport equation with bounded and Dini continuous drift

TL;DR

This paper extends the theory of stochastic transport equations to include bounded and Dini continuous drifts, establishing uniqueness of solutions and stochastic flows, thus broadening the class of drifts for which well-posedness is known.

Contribution

It generalizes previous results from H"older to Dini continuous drifts, proving existence and uniqueness of solutions and stochastic flows under these conditions.

Findings

Established uniqueness of $L^ abla$-solutions with Dini continuous drift.

Proved existence and uniqueness of stochastic diffeomorphism flows.

Partially addressed the open problem of uniqueness with bounded measurable drift.

Abstract

The results established by Flandoli, Gubinelli and Priola ({\it Invent. Math.} {\bf 180} (2010) 1--53) for stochastic transport equation with bounded and H\"{o}lder continuous drift are generalized to bounded and Dini continuous drift. The uniqueness of $L^\infty$-solutions is established by the It\^o--Tanaka trick partially solving the uniqueness problem, which is still open, for stochastic transport equation with only bounded measurable drift. Moreover the existence and uniqueness of stochastic diffeomorphisms flows for a stochastic differential equation with bounded and Dini continuous drift is obtained.