  Consider a system of particles evolving as independent and identically
distributed (i.i.d.) random walks. Initial fluctuations in the particle density
get translated over time with velocity $\vec{v}$, the common mean velocity of
the random walks. Consider a box centered around an observer who starts at the
origin and moves with constant velocity $\vec{v}$. To observe interesting
fluctuations beyond the translation of initial density fluctuations, we measure
the net flux of particles over time into this moving box. We call this the
``box-current" process.
  We generalize this current process to a distribution valued process. Scaling
time by $n$ and space by $\sqrt{n}$ gives current fluctuations of order
$n^{d/4}$ where $d$ is the space dimension. The scaling limit of the normalized
current process is a distribution valued Gaussian process with given
covariance. The limiting current process is equal in distribution to the
solution of a given stochastic partial differential equation which is related
to the generalized Ornstein-Uhlenbeck process.
