Current fluctuations for independent random walks in multiple dimensions

TL;DR

This paper studies the fluctuations of particle flux in a system of independent random walks in multiple dimensions, revealing a Gaussian process limit described by a stochastic PDE.

Contribution

It introduces a distribution-valued current process for independent random walks and characterizes its scaling limit as a Gaussian process solving a specific SPDE.

Findings

Current fluctuations scale as n^{d/4} in d dimensions.

The limit process is a distribution-valued Gaussian process.

The limiting process solves a particular stochastic PDE related to Ornstein-Uhlenbeck.

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.