Renormalized Solutions to Stochastic Continuity Equations with Rough Coefficients

TL;DR

This paper establishes conditions for the renormalization and well-posedness of solutions to stochastic continuity equations with irregular coefficients, providing new proofs of uniqueness without relying on flow regularity.

Contribution

It introduces novel regularity conditions ensuring renormalization and well-posedness for stochastic continuity equations with rough coefficients, avoiding traditional flow regularity assumptions.

Findings

Proved well-posedness in L^p for stochastic continuity equations with irregular coefficients.

Provided a new proof of renormalizability and uniqueness without flow regularity.

Established regularity conditions under which weak solutions are renormalized.

Abstract

We consider the stochastic continuity equation associated to an It\^{o} diffusion with irregular drift and diffusion coefficients. We give regularity conditions under which weak solutions are renormalized in the sense of DiPerna/Lions, and prove well-posedness in $L^p$. As an application, we give a new proof of renormalizability (hence uniqueness) of weak solutions to the stochastic continuity equation when the diffusion matrix is constant and the drift only belongs to $L^q_tL^p$, where $\frac{2}{q} + \frac{n}{p} <1$, without resorting to the regularity of the stochastic flow or a duality method.