New York Journal of Mathematics
Volume 21 (2015) 723-757


Mathav Murugan and Laurent Saloff-Coste

Transition probability estimates for long range random walks

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Published: August 12, 2015
Keywords: long range random walks; heat kernel estimates; heavy-tailed
Subject: 60J10, 60J75, 60J15

Let (M,d,μ) be a uniformly discrete metric measure space satisfying space homogeneous volume doubling condition. We consider discrete time Markov chains on M symmetric with respect to μ and whose one-step transition density is comparable to
(Vh(d(x,y)) ϕ(d(x,y))-1,
where ϕ is a positive continuous regularly varying function with index β ∈ (0,2) and Vh is the homogeneous volume growth function. Extending several existing work by other authors, we prove global upper and lower bounds for n-step transition probability density that are sharp up to constants.


Both authors were partially supported by NSF grants DMS 1004771 and DMS 1404435

Author information

Mathav Murugan:
Department of Mathematics, University of British Columbia and Pacific Institute for the Mathematical Sciences, Vancouver, BC V6T 1Z2, Canada.

Laurent Saloff-Coste:
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA.