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Limit theorem for continuous-time random walks with two time scales

Part of: Markov processes Real functions Limit theorems

Published online by Cambridge University Press:  14 July 2016

Peter Becker-Kern ,
Mark M. Meerschaert  and
Hans-Peter Scheffler
Peter Becker-Kern*
Affiliation:
University of Dortmund
Mark M. Meerschaert*
Affiliation:
University of Nevada
Hans-Peter Scheffler*
Affiliation:
University of Dortmund
*
∗ Postal address: Fachbereich Mathematik, University of Dortmund, D-44221 Dortmund, Germany.
∗∗∗ Postal address: Department of Mathematics, University of Nevada, Reno, NV 89557, USA. Email address: mcubed@unr.edu
∗ Postal address: Fachbereich Mathematik, University of Dortmund, D-44221 Dortmund, Germany.
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Abstract

Continuous-time random walks incorporate a random waiting time between random jumps. They are used in physics to model particle motion. A physically realistic rescaling uses two different time scales for the mean waiting time and the deviation from the mean. This paper derives the scaling limits for such processes. These limit processes are governed by fractional partial differential equations that may be useful in physics. A transfer theorem for weak convergence of finite-dimensional distributions of stochastic processes is also obtained.

Keywords

Anomalous diffusionoperator stable lawcontinuous-time random walkfractional derivative

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles 60J60: Diffusion processes 26A33: Fractional derivatives and integrals
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 455 - 466
Copyright
Copyright © Applied Probability Trust 2004 

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