  We establish a general connection between ballistic and diffusive transport
in systems where the ballistic contribution in canonical ensemble vanishes. A
lower bound on the Green-Kubo diffusion constant is derived in terms of the
curvature of the ideal transport coefficient, the Drude weight, with respect to
the filling parameter. As an application, we explicitly determine the lower
bound on the high temperature diffusion constant in the anisotropic spin 1/2
Heisenberg chain for anisotropy parameters $\Delta \geq 1$, thus settling the
question whether the transport is sub-diffusive or not. Addi- tionally, the
lower bound is shown to saturate the diffusion constant for a certain classical
integrable model.
