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May 2021 Brownian motion on stable looptrees
Eleanor Archer
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Ann. Inst. H. Poincaré Probab. Statist. 57(2): 940-979 (May 2021). DOI: 10.1214/20-AIHP1103

Abstract

In this article, we introduce Brownian motion on α-stable looptrees using resistance techniques, where α(1,2). We prove an invariance principle characterising it as the scaling limit of random walks on discrete looptrees, and prove precise local and global bounds on its heat kernel. We also conduct a detailed investigation of the volume growth properties of stable looptrees, and show that the random volume and heat kernel fluctuations are locally log-logarithmic, and globally logarithmic around leading terms of rα and tαα+1 respectively. These volume fluctuations are the same order as for the Brownian continuum random tree, but the upper volume fluctuations (and corresponding lower heat kernel fluctuations) are different to those of stable trees.

Dans cet article, nous introduisons le mouvement brownien sur les arbres à boucles α-stables en utilisant des techniques de réseaux électriques, pour α(1,2). Nous démontrons un principe d’invariance énonçant que ce processus est la limite de marches aléatoires sur des arbres à boucles discrets, et nous montrons des bornes précises locales et globales sur son noyau de la chaleur. Nous menons également une étude approfondie des propriétés de croissance de volume des arbres à boucles stables, et montrons que les fluctuations aléatoires de volume et du noyau de la chaleur sont localement en logarithme itéré, et globalement logarithmiques, autour d’un terme dominant de rα et tαα+1 respectivement. Ces fluctuations de volume sont du même ordre que pour l’arbre continu aléatoire brownien, mais les bornes supérieures des fluctuations de volume (et les bornes inférieures correspondantes des fluctuations du noyau de la chaleur) sont différentes de celles des arbres stables.

Acknowledgements

I would like to thank my supervisor David Croydon for suggesting the problem and for many helpful discussions, as well as the anonymous referees for their detailed and helpful comments on the initial version of this manuscript. I would also like to thank the Great Britain Sasakawa Foundation for supporting a trip to Kyoto during which some of this work was completed, and Kyoto University for their hospitality during this trip.

Citation

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Eleanor Archer. "Brownian motion on stable looptrees." Ann. Inst. H. Poincaré Probab. Statist. 57 (2) 940 - 979, May 2021. https://doi.org/10.1214/20-AIHP1103

Information

Received: 13 May 2019; Revised: 27 March 2020; Accepted: 1 September 2020; Published: May 2021
First available in Project Euclid: 13 May 2021

Digital Object Identifier: 10.1214/20-AIHP1103

Subjects:
Primary: 54E70, 60F17, 60G52, 60G57, 60K37

Rights: Copyright © 2021 Association des Publications de l’Institut Henri Poincaré

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Vol.57 • No. 2 • May 2021
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