Journal of Theoretical Probability

, Volume 24, Issue 4, pp 1170–1195

# Current Fluctuations for Independent Random Walks in Multiple Dimensions

Open Access
Article

## Abstract

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.

## Keywords

Independent random walks Hydrodynamic limit Current fluctuations Distribution-valued process Generalized Ornstein–Uhlenbeck process

## Mathematics Subject Classification

60K35 60F10 60F17 60G15

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