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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 ∞-solutions are renormalized. But then, if the noise is non-degenerate, uniqueness of weak L ∞-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.