Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Heat kernel estimates for anomalous heavy-tailed random walks

Mathav Murugan and Laurent Saloff-Coste

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Abstract

Sub-Gaussian estimates for the natural random walk is typical of many regular fractal graphs. Subordination shows that there exist heavy tailed jump processes whose jump indices are greater than or equal to two. However, the existing machinery used to prove heat kernel bounds for such heavy tailed random walks fail in this case. In this work we extend Davies’ perturbation method to obtain transition probability bounds for these anomalous heavy tailed random walks. We prove global upper and lower bounds on the transition probability density that are sharp up to constants. An important feature of our work is that the methods we develop are robust to small perturbations of the symmetric jump kernel.

Résumé

Pour de nombreux graphes réguliers de type fractal, la marche aléatoire simple satisfait des estimations de type sous-Gaussiennes. La technique de la subordination montre alors qu’il existe des processus de saut à queue lourde dont l’indice des sauts est supérieur ou égale a 2. Pour de tels processus, les techniques usuelles pour les estimations loin de la diagonale ne fonctionnent pas. Nous étendons la célèbre méthode de Davies dans le cas de ces processus à sauts « anormaux. » Nous obtenons des bornes supérieures et inférieures précises sur le noyau de transition par des méthodes qui sont stables sous de petites perturbations des sauts.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 697-719.

Dates
Received: 27 May 2016
Revised: 4 September 2017
Accepted: 27 February 2018
First available in Project Euclid: 14 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1557820828

Digital Object Identifier
doi:10.1214/18-AIHP895

Mathematical Reviews number (MathSciNet)
MR3949950

Zentralblatt MATH identifier
07097328

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J75: Jump processes

Keywords
Heavy-tailed random walks Jump process Fractals Heat kernel

Citation

Murugan, Mathav; Saloff-Coste, Laurent. Heat kernel estimates for anomalous heavy-tailed random walks. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 697--719. doi:10.1214/18-AIHP895. https://projecteuclid.org/euclid.aihp/1557820828


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