Random walks with infinite spatial and temporal moments

Michael F. Shlesinger, Joseph Klafter, Y. M. Wong

Research output: Contribution to journalArticlepeer-review


The continuous-time random walk of Montroll and Weiss has been modified by Scher and Lax to include a coupled spatial-temporal memory. We treat novel cases for the random walk and the corresponding generalized master equation when combinations of both spatial, and temporal moments of the memory are infinite. The asymptotic properties of the probability distribution for being at any lattice site as a function of time and its variance are calculated. The resulting behavior includes localized, diffusive, wavelike, and Levy's stable laws for the appropriate scaled variable. We show that an infinite mean waiting time can lead to long time diffusive behavior, while a finite mean waiting time is not sufficient to ensure the same.

Original languageEnglish
Pages (from-to)499-512
Number of pages14
JournalJournal of Statistical Physics
Issue number3
StatePublished - Mar 1982
Externally publishedYes


  • Random walk
  • coupled memory
  • infinite moments
  • stable distributions


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