Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-09T16:23:09.224Z Has data issue: false hasContentIssue false
  • English
  • Français

Space–Time Duality for Fractional Diffusion

Part of: Stochastic processes Real functions

Published online by Cambridge University Press:  14 July 2016

Boris Baeumer ,
Mark M. Meerschaert  and
Erkan Nane
Boris Baeumer*
Affiliation:
University of Otago
Mark M. Meerschaert*
Affiliation:
Michigan State University
Erkan Nane*
Affiliation:
Auburn University
*
∗Postal address: Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin, New Zealand. Email address: bbaeumer@maths.otago.ac.nz
∗∗Postal address: Department of Probability and Statistics, Michigan State University, East Lansing, MI 48823, USA. Email address: mcubed@stt.msu.edu
∗∗∗Postal address: 221 Parker Hall, Department of Mathematics and Statistics, Auburn University, Auburn, Al 36849, USA. Email address: nane@auburn.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Zolotarev (1961) proved a duality result that relates stable densities with different indices. In this paper we show how Zolotarev's duality leads to some interesting results on fractional diffusion. Fractional diffusion equations employ fractional derivatives in place of the usual integer-order derivatives. They govern scaling limits of random walk models, with power-law jumps leading to fractional derivatives in space, and power-law waiting times between the jumps leading to fractional derivatives in time. The limit process is a stable Lévy motion that models the jumps, subordinated to an inverse stable process that models the waiting times. Using duality, we relate the density of a spectrally negative stable process with index 1<α<2 to the density of the hitting time of a stable subordinator with index 1/α, and thereby unify some recent results in the literature. These results provide a concrete interpretation of Zolotarev's duality in terms of the fractional diffusion model. They also illuminate a current controversy in hydrology, regarding the appropriate use of space- and time-fractional derivatives to model contaminant transport in river flows.

Keywords

Limit theorystable processsubordinatorfractional diffusionduality

MSC classification

Secondary: 60G52: Stable processes 26A33: Fractional derivatives and integrals
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 4 , December 2009 , pp. 1100 - 1115
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Allouba, H. and Zheng, W. (2001). {Brownian-time processes: the PDE connection and the half-derivative generator}. Ann. Prob. 29, 1780–1795.CrossRefGoogle Scholar
[2] Arendt, W., Batty, C., Hieber, M. and Neubrander, F. (2001). Vector-Valued Laplace transforms and Cauchy Problems (Monogr. Math. 96). Birkhäuser, Basel.CrossRefGoogle Scholar
[3] Baeumer, B. and Meerschaert, M. M. (2001). Stochastic solutions for fractional Cauchy problems. Fractional Calculus Appl. Anal. 4, 481–500.Google Scholar
[4] Baeumer, B., Meerschaert, M. M. and Nane, E. (2009). Brownian subordinators and fractional Cauchy problems. Trans. Amer. Math. Soc. 361, 3915–3930.CrossRefGoogle Scholar
[5] Becker-Kern, P., Meerschaert, M. M. and Scheffler, H. P. (2004). Limit theorems for coupled continuous-time random walks. Ann. Prob. 32, 730–756.CrossRefGoogle Scholar
[6] Benson, D., Wheatcraft, S. and Meerschaert, M. (2000). The fractional-order governing equation of Lévy motion. Water Resources Res. 36, 1413–1424.CrossRefGoogle Scholar
[7] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
[8] Bhattacharya, R. N., Gupta, V. K. and Sposito, G. (1976). On the stochastic foundation of the theory of water flow through unsaturated soil. Water Resources Res. 12, 503–512.CrossRefGoogle Scholar
[9] Bingham, N. H. (1971). Limit theorems for occupation times of Markov processes. Z. Warscheinlichkeitsth. 17, 1–22.CrossRefGoogle Scholar
[10] Bingham, N. H. (1973). Maxima of sums of random variables and suprema of stable processes. Z. Warschein- lichkeitsth. 26, 273–296.Google Scholar
[11] Bondesson, L., Kristiansen, G. K. and Steutel, F. W. (1996). Infinite divisibility of random variables and their integer parts. Statist. Prob. Lett. 28, 271–278.CrossRefGoogle Scholar
[12] Burdzy, K. (1993). {Some path properties of iterated Brownian motion.} In Seminar on Stochastic Processes, eds Çinlar, E., Chung, K. L. and Sharpe, M. J., Birkhäuser, Boston, MA, pp. 67–87.Google Scholar
[13] Caputo, M. (1967). Linear models of dissipation whose Q is almost frequency independent. Part II. Geophys. J. R. Astron. Soc. 13, 529–539.Google Scholar
[14] Chakraborty, P., Meerschaert, M. M. and Lim, C. Y. (2009). Parameter estimation for fractional transport: a particle tracking approach. Water Resources Res. 45, W10415, 15 pp.Google Scholar
[15] Chaves, A. S. (1998). A fractional diffusion equation to describe Lévy flights. Phys. Lett. A 239, 13–16.CrossRefGoogle Scholar
[16] DeBlassie, R. D. (2004). Iterated Brownian motion in an open set. Ann. Appl. Prob. 14, 1529–1558.CrossRefGoogle Scholar
[17] Deng, Z.-Q., Bengtsson, L. and Singh, V. P. (2006). Parameter estimation for fractional dispersion model for rivers. Environm. Fluid Mech. 6, 451–475.Google Scholar
[18] Einstein, A. (1906). On the theory of the Brownian movement. Annalen der Physik 4, 371–381.Google Scholar
[19] Feller, W. (1952). On a generalization of Marcel Riesz' potentials and the semi-groups generated by them. In Comm. Sém. Math. Univ. Lund, Tome Supplementaire, pp. 73–81.Google Scholar
[20] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
[21] Hille, E. and Phillips, R. S. (1957). Functional Analysis and Semi-Groups (Amer. Math. Soc. Coll. Publ. 31). American Mathematical Society, Providence, RI.Google Scholar
[22] Huang, G.-H., Huang, Q.-Z. and Zhan, H.-B. (2006). Evidence of one-dimensional scale-dependent fractional advection-dispersion. J. Contam. Hydrol. 85, 53–71.Google Scholar
[23] Hunt, B. (2006). Asymptotic solutions for one-dimensional dispersion in rivers. J. Hydraulic Eng. 132, 87–93.Google Scholar
[24] Kim, S. and Kavvas, M. L. (2006). Generalized Fick's law and fractional ADE for pollution transport in a river: detailed derivation. J. Hydrol. Eng. 11, 80–83.Google Scholar
[25] Lukacs, E. (1969). Stable distributions and their characteristic functions. Jber. Deutsch Math.-Verein 71, 84–114.Google Scholar
[26] Meerschaert, M. M. and Scheffler, H.-P. (2001). Limit Distributions for Sums of Independent Random Vectors. John Wiley, New York.Google Scholar
[27] Meerschaert, M. M. and Scheffler, H.-P. (2004). Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Prob. 41, 623–638.Google Scholar
[28] Meerschaert, M. M. and Scheffler, H.-P. (2008). Triangular array limits for continuous time random walks. Stoch. Process. Appl. 118, 1606–1633.Google Scholar
[29] Meerschaert, M. M. and Tadjeran, C. (2004). Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77.Google Scholar
[30] Meerschaert, M. M. and Tadjeran, C. (2006). Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56, 80–90.CrossRefGoogle Scholar
[31] Meerschaert, M. M., Nane, E. and Xiao, Y. (2008). Large deviations for local time fractional Brownian motion and applications. J. Math. Anal. Appl. 346, 432–445.Google Scholar
[32] Meerschaert, M. M., Nane, E. and Vellaisamy, P. (2009). Fractional Cauchy problems on bounded domains. Ann. Prob. 37, 979–1007.Google Scholar
[33] 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
[34] Metzler, R. and Klafter, J. (2004). The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A 37, R161–R208.CrossRefGoogle Scholar
[35] Miller, K. S. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley, New York.Google Scholar
[36] Montroll, E. W. and Weiss, G. H. (1965). Random walks on lattices. II. J. Math. Phys. 6, 167–181.Google Scholar
[37] Nigmatullin, R. R. (1986). The realization of the generalized transfer in a medium with fractal geometry. Phys. Status Solidi B 133, 425–430.Google Scholar
[38] Nolan, J. P. (1997). Numerical calculation of stable densities and distribution functions. Heavy tails and highly volatile phenomena. Commun. Statist. Stoch. Models 13, 759–774.Google Scholar
[39] Orsingher, E. and Beghin, L. (2004). Time-fractional telegraph equations and telegraph processes with Brownian time. Prob. Theory. Relat. Fields 128, 141–160.Google Scholar
[40] Orsingher, E. and Beghin, L. (2009). Fractional diffusion equations and processes with randomly varying time. Ann. Prob. 37, 206–249.Google Scholar
[41] Podlubny, I. (1999). Fractional Differential Equations. Academic Press, San Diego, CA.Google Scholar
[42] Riesz, M. (1949). L{'}intégrale de Riemann–Liouville et le probléme de Cauchy. Acta Math. 81, 1–223.Google Scholar
[43] Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives. Gordon and Breach, Yverdon.Google Scholar
[44] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Processes. Chapman and Hall, New York.Google Scholar
[45] Scalas, E. (2004). Five years of continuous-time random walks in econophysics. In Proc. WEHIA (Kyoto, 2004), ed. Namatame, A., Springer, Berlin, pp. 3–16.Google Scholar
[46] Scher, H. and Lax, M. (1973). Stochastic transport in a disordered solid. I. Theory. Phys. Rev. B 7, 4491–4502.Google Scholar
[47] Schumer, R., Meerschaert, M. M. and Baeumer, B. (2009). Fractional advection-dispersion equations for modeling transport at the Earth surface. To appear in J. Geophys. Res. Available at www.stt.msu.edu/∼mcubed/fADEreview.pdf.Google Scholar
[48] Zaslavsky, G. M. (1994). Fractional kinetic equation for Hamiltonian chaos. Chaotic advection, tracer dynamics and turbulent dispersion. Phys. D 76, 110–122.Google Scholar
[49] Zhang, Y., Benson, D. A. and Reeves, D. M. (2009). Time and space nonlocalities underlying fractional-derivative models: distinction and literature review of field applications. Adv. Water Resources 32, 561–581.CrossRefGoogle Scholar
[50] Zhang, Y., Meerschaert, M. M. and Baeumer, B. (2008). Particle tracking for time-fractional diffusion. Phys. Rev. E 78, 036705.Google Scholar
[51] Zolotarev, V. M. (1961). Expression of the density of a stable distribution with exponent α greater than one by means of a density with exponent 1/α. Dokl. Akad. Nauk SSSR 98, 735–738 (in Russian). English translation: Select. Transl. Math. Statist. Prob. 1, 163-167.Google Scholar