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We investigate the effects of the propagation of a non-degenerate Browniannoise through a chain of deterministic differential equations whosecoefficients are rough and satisfy a weak like H{\"o}rmander structure (i.e. anon-degeneracy condition w.r.t. the components which transmit the noise). Inparticular we characterize, through suitable counterexamples , almost sharpregularity exponents that ensure that weak well posedness holds for theassociated SDE. As a by-product of our approach, we also derive some densityestimates of Krylov type for the weak solutions of the considered SDEs.

Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result