W^{1,p}-solutions of the transport equation by stochastic perturbation

TL;DR

This paper establishes the well-posedness of stochastic transport equations with unbounded H"older continuous vector fields, proving existence, uniqueness, and stability of solutions in W^{1,p} spaces.

Contribution

It introduces a novel approach to prove well-posedness for stochastic transport equations with unbounded H"older continuous vector fields.

Findings

Existence of W^{1,p}-weak solutions.

Uniqueness of solutions under given conditions.

Strong stability of solutions.

Abstract

We consider the stochastic transport equation with a possibly unbounded H\"older continuous vector field. Well-posedness is proved, namely, we show existence, uniqueness and strong stability of W^{1,p}-weak solutions.