Current Fluctuations for Independent Random Walks in Multiple Dimensions

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 (v) over right arrow, 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 (v) over right arrow. 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 root 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.