Stochastic transport equation with singular drift

TL;DR

This paper establishes existence, uniqueness, and regularity of solutions to a stochastic transport equation with highly singular drift fields, extending the class of vector fields for which well-posedness is known.

Contribution

It proves well-posedness and regularity results for stochastic transport equations with very singular drifts, including those not in traditional local Lebesgue spaces.

Findings

Existence and uniqueness of weak solutions.

Sobolev regularity of solutions.

Applicable to drifts in broad singular classes.

Abstract

We prove existence, uniqueness and Sobolev regularity of weak solution of the Cauchy problem of the stochastic transport equation with drift in a large class of singular vector fields containing, in particular, the $L^d$ class, the weak $L^d$ class, as well as some vector fields that are not even in $L_{\rm loc}^{2+\varepsilon}$ for any $\varepsilon>0$.