Stochastic transport equations with unbounded divergence

TL;DR

This paper investigates the existence and uniqueness of solutions to stochastic transport equations with irregular coefficients and unbounded divergence, highlighting the role of smoothing as a selection criterion.

Contribution

It establishes new results on solution existence and uniqueness under less restrictive conditions on divergence and demonstrates smoothing as a selection mechanism.

Findings

Existence and uniqueness proven for equations with irregular coefficients.

Smoothing acts as a selection criterion without divergence restrictions.

Results extend understanding of stochastic transport equations with unbounded divergence.

Abstract

We study in this article the existence and uniqueness of solutions to a class of stochastic transport equations with irregular coefficients and unbounded divergence. In the first result we assume the drift is $L^{2}([0,T] \times \R^{d})\cap L^{\infty}([0,T] \times \R^{d})$ and the divergence is the locally integrable. In the second result we show that the smoothing acts as a selection criterion when the drift is in $L^{2}([0,T] \times \R^{d})\cap L^{\infty}([0,T] \times \R^{d})$ without any condition on the divergence.