Stochastic transport equation in bounded domains

TL;DR

This paper investigates the stochastic transport equation in bounded domains, establishing existence and uniqueness of weak solutions under certain conditions, and highlighting the importance of stochastic trace and formulation conversion.

Contribution

It introduces new results on existence and uniqueness of weak solutions for stochastic transport equations with non-regular coefficients in bounded domains.

Findings

Proved existence of weak solutions using stochastic trace.

Established uniqueness of solutions under specific conditions.

Analyzed the role of Stratonovich to Itô conversion in bounded domains.

Abstract

This paper is concerned with the initial-boundary value problem \; for stochastic transport equations in bounded domains. For a given stochastic perturbation of the drift vector field, we prove existence and uniqueness of weak solutions with non-regular coefficients. The existence of weak solutions relies strongly on the existence of strong stochastic trace, and also on the passage from the Stratonovich into It\^o's formulation for bounded domains.Moreover, the uniqueness of weak solutions is obtained under suitable conditions.