On massive sets for subordinated random walks

We study massive (reccurent) sets with respect to a certain random walk Sα defined on the integer lattice Zd, d=1,2. Our random walk Sα is obtained from the simple random walk S on Zd by the procedure of discrete subordination. Sα can be regarded as a discrete space and time counterpart of the symme...

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Bibliographic Details
Published in:Mathematische Nachrichten 2015-06, Vol.288 (8-9), p.841-853
Main Authors: Bendikov, Alexander, Cygan, Wojciech
Format: Article
Language:English
Subjects:
05C81
31A15
60J45
Capacity
Green function
random walk
regular variation
subordination
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Summary:We study massive (reccurent) sets with respect to a certain random walk Sα defined on the integer lattice Zd, d=1,2. Our random walk Sα is obtained from the simple random walk S on Zd by the procedure of discrete subordination. Sα can be regarded as a discrete space and time counterpart of the symmetric α‐stable Lévy process in Rd. In the case d=1 we show that some remarkable proper subsets of Z , e.g. the set P of primes, are massive whereas some proper subsets of P such as the Leitmann primes Ph are massive/non‐massive depending on the function h. Our results can be regarded as an extension of the results of McKean (1961) about massiveness of the set of primes for the simple random walk in Z3. In the case d=2 we study massiveness of thorns and their proper subsets. The case d>2 is presented in the recent paper Bendikov and Cygan .
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.201400037