  Consider a tagged particle in zero-range dynamics on the integer lattice in
dimension d with rate g whose finite-range jump probabilities p possess a
drift. We show, in equilibrium, that the variance of the tagged particle
position at time t is at least order t in all dimensions and at most order t in
d=1 and d larger or equal to 3 for a wide class of rates g. Also, in d=1, when
the jump distribution p is totally asymmetric and nearest-neighbor, and when
the rate g(k) increases and g(k)/k decreases with k, we show the diffusively
scaled centered tagged particle position converges to a Brownian motion.
