$L^{p}-$solutions of the stochastic transport equation

TL;DR

This paper establishes existence, uniqueness, and representation of weak solutions for the stochastic transport equation, extending classical results with minimal assumptions and a Wong-Zakai approximation.

Contribution

It introduces a new framework for $L^{p}$-solutions of stochastic transport equations, combining Itô-Ventzel-Kunita formula and Lions-DiPerna theory.

Findings

Proved existence and uniqueness of weak $L^{p}$-solutions.

Provided a solution representation and Wong-Zakai approximation.

Extended classical deterministic results to stochastic setting.

Abstract

We consider the stochastic transport linear equation and we prove existence and uniqueness of weak $L^{p}-$solutions. Moreover, we obtain a representation of the general solution and a Wong-Zakai principle for this equation. We make only minimal assumptions, similar to the deterministic problem. The proof is supported on the generalized It\^o-Ventzel-Kunita formula (see 15) and the theory of Lions-DiPerna on transport linear equation (see 9).