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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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.


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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Heavy-tailed random walks Jump process Fractals Heat kernel


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.

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