Quadratic Fluctuations of the Simple Exclusion Process

TL;DR

This paper introduces a quadratic field linked to the Ornstein-Uhlenbeck process and demonstrates that the rescaled quadratic fluctuations of the simple exclusion process converge to this field, revealing new insights into its properties.

Contribution

The paper defines a new quadratic field and proves the convergence of the simple exclusion process's fluctuations to this field under diffusive scaling.

Findings

Convergence of fluctuations to the quadratic field

Representation of the quadratic field as Wick-renormalized square

Analysis of small and large-time properties of the quadratic field

Abstract

We introduce a two-dimensional, distribution-valued field which we call the quadratic field associated to the one-dimensional Ornstein-Uhlenbeck process. We show that the stationary quadratic fluctuations of the simple exclusion process, when rescaled in the diffusive scaling, converge to this quadratic field. We show that this quadratic field evaluated at the diagonal corresponds to the Wick-renormalized square of the Ornstein-Uhlenbeck process, and we use this new representation in order to prove some small and large-time properties of it.