Comptes Rendus
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Singularity of the loops within a cable-graph loop-soup conditioned by its occupation time
[Singularité des boucles d’une soupe de boucles browniennes conditionnée par son temps d’occupation sur un graphe métrique]
Arthur Dremaux1
1University of Cambridge, United Kingdom
Comptes Rendus. Mathématique, Volume 364 (2026), pp. 311-319

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.

Nous montrons la propriété suivante de la relation entre les soupes de boucles browniennes sur un graphe métrique et leur profil de temps d’occupation $\Lambda $ : quand on conditionne sur $\Lambda $, la loi conditionnée des boucles individuelles devient singulière par rapport à celle des boucles non-conditionnées. L’idée de la preuve est de remarquer qu’un certain type de « points rapides » sur la courbe $\Lambda $ impose un comportement exceptionnel à toutes les boucles lorsqu’elles passent par ces points.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.827
Classification : 60J65
Keywords: Brownian loop-soup, cable-graph, local time, fast points, squared Bessel processes, Brownian burglar
Mots-clés : Soupe de boucles browniennes, graphe métrique, temps local, carrés de processus de Bessel, voleur brownien, points rapides

Arthur Dremaux  1

1 University of Cambridge, United Kingdom
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Arthur Dremaux. Singularity of the loops within a cable-graph loop-soup conditioned by its occupation time. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 311-319. doi: 10.5802/crmath.827
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