On the occupation times of Brownian excursions and Brownian loops

TL;DR

This paper investigates the occupation times of Brownian excursions and loops in 2D domains, linking them to Gaussian fields like the Gaussian Free Field through normalization and scaling limits.

Contribution

It introduces a novel interpretation of Gaussian Free Fields as fluctuations of occupation times of Brownian excursion clouds in the high-intensity limit.

Findings

Gaussian Free Fields can be viewed as occupation time fluctuations

Occupation times of Brownian excursions relate to Gaussian fields

Provides a framework for understanding Gaussian fields via Brownian loop and excursion measures

Abstract

We study properties of occupation times by Brownian excursions and Brownian loops in two-dimensional domains. This allows for instance to interpret some Gaussian fields, such as the Gaussian Free Fields as (properly normalized) fluctuations of the total occupation time of a Poisson cloud of Brownian excursions when the intensity of the cloud goes to infinity.