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In this talk I will present a reducibility result holding for time quasi-periodic unbounded perturbations of a non-resonant transport equation on the dâ dimensional torus $T^d$ . Such a result is obtained combining pseudo-differential calculus, used to conjugate the initial system to a new one with a smoothing perturbation, and a KAM scheme, implemented to actually obtain reducibility. This makes possible to exhibit an example of reducibility for a higher dimensional Hamiltonian PDE, in a case where, in addition, the unperturbed problem has a spectrum which is dense on the real axis.