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

Part of: Real functions Markov processes 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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References

Baeumer, B., and Meerschaert, M. (2001). Stochastic solutions for fractional Cauchy problems. Fractional Calculus Appl. Anal. 4, 481–500.Google Scholar
Baeumer, B., Benson, D. A., and Meerschaert, M. M. (2003). Advection and dispersion in time and space. Preprint, University of Nevada. Available at http://unr.edu/homepage/mcubed/.Google Scholar
Baxter, G., and Donsker, M. D. (1957). On the distribution of the supremum functional for processes with stationary independent increments. Trans. Amer. Math. Soc. 85, 73–87.CrossRefGoogle Scholar
Caputo, M. (1967). Linear models of dissipation whose Q is almost frequency independent. Part II. Geophys. J. R. Astr. Soc. 13, 529–539.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Jurek, Z., and Mason, J. D. (1993). Operator-Limit Distributions in Probability Theory. John Wiley, New York.Google Scholar
Haggerty, R., McKenna, S. A., and Meigs, L. C. (2000). On the late-time behavior of tracer test breakthrough curves. Water Resources Res. 36, 3467–3479.Google Scholar
Meerschaert, M., and Scheffler, H. P. (2001). Limit Distributions for Sums of Independent Random Vectors. John Wiley, New York.Google Scholar
Meerschaert, M., Benson, D., and Baeumer, B. (1999). Multidimensional advection and fractional dispersion. Phys. Rev. E 59, 5026–5028.Google Scholar
Meerschaert, M., Benson, D., and Baeumer, B. (2001). Operator Lévy motion and multiscaling anomalous diffusion. Phys. Rev. E 63, 1112–1117.Google Scholar
Meerschaert, M. M., and Scheffler, H. P. (2001). Limit theorems for continuous time random walks. Preprint, University of Nevada. Available at http://unr.edu/homepage/mcubed/.Google Scholar
Meerschaert, M. M., Benson, D. A., Scheffler, H. P., and Baeumer, B. (2002). Stochastic solution of space-time fractional diffusion equations. Phys. Rev. E 65, 1103–1106.Google Scholar
Podlubny, I. (1999). Fractional Differential Equations. Academic Press, San Diego, CA.Google Scholar
Ross, S. (1993). Introduction to Probability Models, 5th edn. Academic Press, Boston, MA.Google Scholar
Sato, K. I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Operat. Res. 5, 67–85.Google Scholar
Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.Google Scholar
Zaslavsky, G. (1994). Fractional kinetic equation for Hamiltonian chaos. Chaotic advection, tracer dynamics and turbulent dispersion. Phys. D 76, 110–122.Google Scholar