The law of a point process of Brownian excursions in a domain is determined by the law of its trace

TL;DR

This paper proves that the distribution of a point process of Brownian excursions within a domain is uniquely determined by the distribution of its trace, impacting the understanding of Brownian loop-soup clusters in two dimensions.

Contribution

It establishes that the law of a point process of Brownian excursions is fully determined by its trace, providing new insights into Brownian loop-soup cluster decomposition.

Findings

The law of the point process is determined by its trace.

Implications for Brownian loop-soup cluster decomposition in 2D.

Theoretical foundation for analyzing Brownian excursions and loops.

Abstract

We show the result that is stated in the title of the paper, which has consequences about decomposition of Brownian loop-soup clusters in two dimensions.