Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scaling limits of stochastic processes associated with resistance forms

D. A. Croydon

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Abstract

We establish that if a sequence of spaces equipped with resistance metrics and measures converge with respect to the Gromov–Hausdorff-vague topology, and a certain non-explosion condition is satisfied, then the associated stochastic processes also converge. This result generalises previous work on trees, fractals, and various models of random graphs. We further conjecture that it will be applicable to the random walk on the incipient infinite cluster of critical bond percolation on the high-dimensional integer lattice.

Résumé

Nous établissons que si une suite d’espaces équipée des métriques de résistance et de mesures converge par rapport à la topologie de Gromov–Hausdorff-vague, et qu’une certaine condition de non explosion est satisfaite, alors les processus stochastiques associés convergent également. Ces résultats généralisent des travaux précédents sur les arbres, fractals et divers modèles de graphes aléatoires. De plus nous conjecturons que cela devrait s’appliquer à la marche aléatoire sur l’amas de percolation par arêtes au point critique conditionné à être infini sur les réseaux entiers de grande dimension.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 4 (2018), 1939-1968.

Dates
Received: 23 January 2017
Revised: 29 May 2017
Accepted: 21 August 2017
First available in Project Euclid: 18 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1539849789

Digital Object Identifier
doi:10.1214/17-AIHP861

Mathematical Reviews number (MathSciNet)
MR3865663

Zentralblatt MATH identifier
06996555

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 28A80: Fractals [See also 37Fxx] 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Keywords
Fractal Gromov–Hausdorff-vague topology Random graph Resistance form Resolvent kernel Tree

Citation

Croydon, D. A. Scaling limits of stochastic processes associated with resistance forms. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 4, 1939--1968. doi:10.1214/17-AIHP861. https://projecteuclid.org/euclid.aihp/1539849789


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