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January 2022 Scaling limit of the subdiffusive random walk on a Galton–Watson tree in random environment
Loïc de Raphélis
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Ann. Probab. 50(1): 339-396 (January 2022). DOI: 10.1214/21-AOP1535

Abstract

We consider a random walk on a Galton–Watson tree in random environment, in the subdiffusive case. We prove the convergence of the renormalised height function of the walk towards the continuous-time height process of a spectrally positive strictly stable Lévy process, jointly with the convergence of the renormalised trace of the walk towards the real tree coded by the latter continuous-time height process.

Funding Statement

Partially supported by the ANR project Liouville (ANR-15-CE40-0013).

Acknowledgements

I thank my advisor Elie Aïdékon for guiding me throughout the development of this article. I thank Xinxin Chen for her advice on the organization of the proofs and for spotting mistakes in an earlier version. I also thank an anonymous referee for his/her numerous precise comments which greatly improved the quality of this paper.

Citation

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Loïc de Raphélis. "Scaling limit of the subdiffusive random walk on a Galton–Watson tree in random environment." Ann. Probab. 50 (1) 339 - 396, January 2022. https://doi.org/10.1214/21-AOP1535

Information

Received: 1 April 2019; Revised: 1 April 2021; Published: January 2022
First available in Project Euclid: 23 February 2022

Digital Object Identifier: 10.1214/21-AOP1535

Subjects:
Primary: 60F17 , 60G50 , 60J80 , 60K37

Keywords: Galton–Watson tree , random environment , Random walk , Scaling limit , stable tree

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 1 • January 2022
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