  We study the equilibrium fluctuations of a tagged particle in finite-range
simple exclusion processes on Z^d with biased single particle jump rates. It is
known the variance of the tagged particle at time t is diffusive, that is on
order O(t), in d\geq 3, and in d=1 when in addition the jump rate is
nearest-neighbor, and moreover, in these cases, central limit theorems in
diffusive scale have been proved.
  In this article, we give some partial results in the open cases in d\leq 2.
Namely, we show diffusivity of the tagged particle variance at time t in the
sense of some upper and lower bounds on order O(t) in d=2, and also in d=1 when
in addition the jump rate is not nearest-neighbor. Also, a characterization of
the tagged particle variance is given. The main methods are in analyzing H_{-1}
norm variational inequalities.
