Renormalized solutions for stochastic transport equations and the regularization by bilinear multiplicative noise

TL;DR

This paper demonstrates that bilinear multiplicative noise can regularize solutions of stochastic transport equations with irregular coefficients, ensuring uniqueness without additional assumptions, unlike in the deterministic case.

Contribution

It shows that nondegenerate bilinear multiplicative noise ensures uniqueness of weak solutions without extra conditions, providing a new perspective on noise-induced regularization.

Findings

All weak L-infinity solutions are renormalized under the same assumptions as deterministic theory.

Nondegenerate noise guarantees uniqueness without requiring bounded divergence of the drift.

The proof offers a new explanation for the regularizing effect of bilinear multiplicative noise.

Abstract

A linear stochastic transport equation with non-regular coefficients is considered. Under the same assumption of the deterministic theory, all weak $L^\infty$-solutions are renormalized. But then, if the noise is nondegenerate, uniqueness of weak $L^\infty$-solutions does not require essential new assumptions, opposite to the deterministic case where for instance the divergence of the drift is asked to be bounded. The proof gives a new explanation why bilinear multiplicative noise may have a regularizing effect.