Singularity of the loops within a cable-graph loop-soup conditioned by its occupation time

TL;DR

This paper demonstrates that conditioning Brownian loop-soups on a cable-graph on their occupation time causes the loops' law to become singular, due to special points influencing their behavior.

Contribution

It reveals a novel singularity property of loop measures conditioned on occupation times, highlighting the impact of fast points on loop behavior.

Findings

Conditional law of loops becomes singular when conditioned on occupation time.

Fast points on the occupation curve impose exceptional behavior on loops.

The proof involves analyzing the influence of special points on loop trajectories.

Abstract

In this note, we show the following feature of the relation between Brownian loop-soups on cable-graphs and their total occupation time-field $\Lambda$: When conditioned on $\Lambda$, the conditional law of individual loops becomes singular with respect to that of unconditioned loops. The idea of the proof is to see that some type of fast points on the curve $\Lambda$ impose an exceptional behaviour of all the loops when they go through these points.