Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-25T22:57:22.238Z Has data issue: false hasContentIssue false
  • English
  • Français

Inverse Stable Subordinators

Published online by Cambridge University Press:  24 April 2013

M. M. Meerschaert  and
P. Straka
M. M. Meerschaert
Affiliation:
Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824
P. Straka*
Affiliation:
School of Mathematics, University of Manchester, Manchester, M13 9PL, United Kingdom
*
⋆ Corresponding author. E-mail: [email protected]
Get access

Abstract

The inverse stable subordinator provides a probability model for time-fractional differential equations, and leads to explicit solution formulae. This paper reviews properties of the inverse stable subordinator, and applications to a variety of problems in mathematics and physics. Several different governing equations for the inverse stable subordinator have been proposed in the literature. This paper also shows how these equations can be reconciled.

Keywords

fractional derivativesubordinatorgoverning equationhitting time
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 2: Anomalous diffusion , 2013 , pp. 1 - 16
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

W. Arendt, C. J. K. Batty, M. Hieber,F. Neubrander. Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics 96, Birkhäuser Verlag, Basel, 2001.
Baeumer, B., Meerschaert, M. M.. Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4 (2001), 481–500. Google Scholar
Barkai, E., Metzler, R., Klafter, J.. From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E 61 (2000), 132. CrossRefGoogle Scholar
Benson, D.A., Meerschaert, M.M.. A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations. Adv. Water Resour. 32 (2009), 532–539. CrossRefGoogle Scholar
Berkowitz, B., Cortis, A., Dentz, M., Scher, H.. Modeling non-Fickian transport in geological formations as a continuous time random walk. Rev. Geophys. 44 (2006), 1–49. CrossRefGoogle Scholar
Bingham, N. H.. Limit theorems for occupation times of Markov processes. Z. Wahrsch. verw. Geb. 17 (1971), 1–22. CrossRefGoogle Scholar
R. Durrett. Probability: Theory and Examples. Cambridge University Press, New York, 2010.
Einstein, A.. On the movement of small particles suspended in a stationary liquid demanded by the molecular kinetic theory of heat. Ann. Phys. 17 (1905), 549–560. CrossRefGoogle Scholar
W. Feller. An Introduction to Probability Theory and Its Applications. 2nd ed., Wiley, New York, 1971.
Hahn, M. G, Kobayashi, K., Umarov, S.. Fokker-Planck-Kolmogorov equations associated with time-changed fractional Brownian motion. Proc. Amer. Math. Soc., 139 (2011), 691–705. CrossRefGoogle Scholar
N. Jacob. Pseudo differential operators and Markov processes. Vol. I. Imperial College Press, London, 2001.
Kochubei, A. N.. A Cauchy problem for evolution equations of fractional order. Diff. Eq. 25 (1989), 967–974. Google Scholar
Kochubei, A. N.. Fractional-order diffusion. Diff. Eq. 26 (1990), 485–492. Google Scholar
Kolokoltsov, V. N.. Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics. Th. Prob. Appl. 53 (2009), 594. CrossRefGoogle Scholar
R.L. Magin. Fractional Calculus in Bioengineering. Begell House, 2006.
Magdziarz, M., Weron, A.. Competition between subdiffusion and Lévy flights: Stochastic and numerical approach. Phys. Rev. E 75 (2007), 056702. CrossRefGoogle Scholar
F. Mainardi. Fractals and fractional calculus in continuum mechanics. Springer Verlag, 1997.
Mainardi, F., Gorenflo, R.. On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118 (2000), 283–299. CrossRefGoogle Scholar
Masoliver, J., Montero, M., Perelló, J., Weiss, G.H., Perello, J.. The continuous time random walk formalism in financial markets. J. Econ. Behav. Org. 61 (2006), 577–598. CrossRefGoogle Scholar
Metzler, R., Klafter, J.. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1–77. CrossRefGoogle Scholar
Metzler, R., Klafter, J.. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37 (2004), R161–R208. CrossRefGoogle Scholar
M. M. Meerschaert, H.-P. Scheffler. Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice. Wiley, New York, 2001.
Meerschaert, M. M., Scheffler, H.-P., Kern, P.. Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Probab. 41 (2004), 455–466. CrossRefGoogle Scholar
Meerschaert, M. M., Scheffler, H.-P.. Triangular array limits for continuous time random walks. Stoch. Proc. Appl. 118 (2008), 1606–1633. CrossRefGoogle Scholar
Meerschaert, M. M., Nane, E., Vellaisamy, P.. Fractional Cauchy problems on bounded domains. Ann. Probab. 37 (2009), 979–1007. CrossRefGoogle Scholar
M. M. Meerschaert, A. Sikorskii. Stochastic Models for Fractional Calculus. De Gruyter, Berlin, 2012.
Metzler, R., Barkai, E., Klafter, J.. Deriving fractional Fokker-Planck equations from a generalised master equation. Europhys. Lett. 46 (1999), 431–436. CrossRefGoogle Scholar
Montroll, E., Weiss, G.. Random walks on lattices. II. J. Math Phys. 6 (1965), 167–181. CrossRefGoogle Scholar
Nigmatullin, R. R.. The realization of the generalized transfer in a medium with fractal geometry. Phys. Status Solidi B 133 (1986), 425–430. CrossRefGoogle Scholar
A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer-Verlag, New York, 1983.
Piryatinska, A., Saichev, A. I., Woyczynski, W.. Models of anomalous diffusion: The subdiffusive case. Phys. A 349 (2005), 375–420. CrossRefGoogle Scholar
I. Podlubny. Fractional differential equations, Academic press, 1999.
R Development Core Team. R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2010.
J. Sabatier, O.P. Agrawal, J.A.T. Machado. Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, 2007.
S. G. Samko, A. A. Kilbas, O. I. Marichev. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, London, 1993.
Scalas, E.. Five years of continuous-time random walks in econophysics. Complex Netw. Econ. Inter. 567 (2006), 3–16. CrossRefGoogle Scholar
A.I. Saichev, W.A. Woyzczynski. Distributions in the Physical and Engineering Sciences: Distributional and Fractal Calculus, Integral Transforms, and Wavelets. Birkhäuser, 1997.
Saichev, A.I., Zaslavsky, G.M.. Fractional kinetic equations: solutions and applications. Chaos, 7 (1997), 753–764. CrossRefGoogle ScholarPubMed
Scher, H., Lax, M.. Stochastic transport in a disordered solid. I. Theory. Phys. Rev. B 7 (1973), 4491–4502. CrossRefGoogle Scholar
Schneider, W. R., Wyss, W.. Fractional diffusion and wave equations. J. Math. Phys. 30 (1989), 134–144. CrossRefGoogle Scholar
I. N. Sneddon. Fourier Transforms. Dover, New York, 1995.
I. Stakgold, M. J. Holst. Green’s functions and boundary value problems. Wiley, New York, 1998.
V. V. Uchaikin, V. M. Zolotarev. Chance and Stability. Stable Distributions and Their Applications. VSP, Utrecht, 1999.
Zaslavsky, G.. Fractional kinetic equation for Hamiltonian chaos. Phys. D 76 (1994), 110–122. CrossRefGoogle Scholar
Zhang, Y., Benson, D., Meerschaert, M. M., Scheffler, H.. On using random walks to solve the space-fractional advection-dispersion equations. J. Stat. Phys. 123 (2006), 89–110. Google Scholar
V. Zolotarev. One-dimensional Stable Distributions. Translations of Mathematical Monographs 65, American Mathematical Society, Providence, RI, 1986.