We introduce a new diffusion mechanism from the neighborhood of elliptic equilibria for Hamiltonian flows in three or more degrees of freedom. Using this mechanism, we obtain the first examples of real analytic Hamiltonians that have a Lyapunov unstable non-resonant elliptic equilibrium. In four or more degrees of freedom, we obtain examples of unstable elliptic equilibria with arbitrary chosen frequency vectors whose coordinates are not all of the same sign. Moreover, the Birkhoff normal form at the origin is divergent in all our examples. In addition, it is possible to insure a transversality condition at the equilibria and the diffusion coexists therefore with the stability in a probabilistic sense (or KAM stability) of the equilibrium.