Random walks with infinite spatial and temporal moments |
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Authors: | Michael F. Shlesinger Joseph Klafter Y. M. Wong |
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Affiliation: | (1) Institute for Physical Science and Technology, University of Maryland, 20742 College Park, Maryland;(2) La Jolla Institute, 92038 La Jolla, California;(3) Exxon Research and Engineering Company, 07036 Linden, New Jersey;(4) Department of Physics, Catholic University, 20064 Washington, D.C. |
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Abstract: | 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. |
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Keywords: | Random walk coupled memory infinite moments stable distributions |
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