On diffusivity of a tagged particle in asymmetric zero-range dynamics

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.