Zero-noise selection and Large Deviations in $L_t^{\mathrm{\infty }}{L}_x^p$ for the stochastic transport equation beyond DiPerna-Lions

We consider $L_t^{\mathrm{\infty }}{L}_x^p$ solutions of the stochastic transport equation with drift in $L_t^{\mathrm{\infty }}{W}_x^{1,q}$. We show strong existence and pathwise uniqueness of solutions in a regime of parameters $p,q$ for which non-unique weak solutions of the deterministic transport equation exist. When the intensity of the noise goes to zero, we prove that the solutions of the stochastic transport equation converge to the unique renormalized solution of the transport equation in the sense of DiPerna-Lions. Furthermore, we show that the convergence is governed by a Large Deviations Principle in the space $L_t^{\mathrm{\infty }}{L}_x^p$. Since the space $L_t^{\mathrm{\infty }}{L}_x^p$ is not separable, the weak convergence approach to Large Deviations by Budhiraja, Dupuis, and Maroulas is not directly applicable.