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

A linear stochastic transport equation with non-regular coefficients is considered. Under the same assumption of the deterministic theory, all weak $L^{\mathrm{\infty }}$-solutions are renormalized. But then, if the noise is nondegenerate, uniqueness of weak $L^{\mathrm{\infty }}$-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.