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Fractional kinetic equations driven by Gaussian or infinitely divisible noise

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  01 July 2016

J. M. Angulo ,
V. V. Anh ,
R. McVinish  and
M. D. Ruiz-Medina
J. M. Angulo*
Affiliation:
University of Granada
V. V. Anh*
Affiliation:
Queensland University of Technology
R. McVinish*
Affiliation:
Queensland University of Technology
M. D. Ruiz-Medina*
Affiliation:
University of Granada
*
∗ Postal address: Department of Statistics and Operations Research, University of Granada, Campus Fuente Nueva s/n, E-18071 Granada, Spain.
∗∗∗ Postal address: School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, QLD 4001, Australia.
∗∗∗ Postal address: School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, QLD 4001, Australia.
∗ Postal address: Department of Statistics and Operations Research, University of Granada, Campus Fuente Nueva s/n, E-18071 Granada, Spain.
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Abstract

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In this paper, we consider a certain type of space- and time-fractional kinetic equation with Gaussian or infinitely divisible noise input. The solutions to the equation are provided in the cases of both bounded and unbounded domains, in conjunction with bounds for the variances of the increments. The role of each of the parameters in the equation is investigated with respect to second- and higher-order properties. In particular, it is shown that long-range dependence may arise in the temporal solution under certain conditions on the spatial operators.

Keywords

Fractional diffusionfractional heat equationfractional kinetic equationlong-range dependence

MSC classification

Primary: 60G60: Random fields
Secondary: 60J60: Diffusion processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 2 , June 2005 , pp. 366 - 392
Copyright
Copyright © Applied Probability Trust 2005 

References

Adler, R. J. (1981). The Geometry of Random Fields. John Wiley, Chichester.Google Scholar
Angulo, J. M., Ruiz-Medina, M. D., Anh, V. V. and Grecksch, W. (2000). Fractional diffusion and fractional heat equation. Adv. Appl. Prob. 32, 1077–1099.Google Scholar
Anh, V. V. and Leonenko, N. N. (2001). Spectral analysis of fractional kinetic equations with random data. J. Statist. Phys. 104, 1349–1387.CrossRefGoogle Scholar
Anh, V. V. and Leonenko, N. N. (2002). Renormalization and homogenization of fractional diffusion equations with random data. Prob. Theory Relat. Fields 124, 381–408.CrossRefGoogle Scholar
Anh, V. V. and Leonenko, N. N. (2003). Harmonic analysis of random fractional diffusion–wave equations. J. Appl. Math. Comput. 141, 77–85.CrossRefGoogle Scholar
Anh, V. V. and McVinish, R. (2003). Complete monotonicity of fractional Green functions with applications. Fractional Calculus Appl. Anal. 6, 157–173.Google Scholar
Anh, V. V. and McVinish, R. (2004). The Riesz–Bessel fractional diffusion equation. Appl. Math. Optimization 49, 241–264.CrossRefGoogle Scholar
Anh, V. V., Heyde, C. C. and Leonenko, N. N. (2002). Dynamic models of long-memory processes driven by Lévy noise. J. Appl. Prob. 39, 730–747.Google Scholar
Anh, V. V., Leonenko, N. N. and McVinish, R. (2001). Models for fractional Riesz–Bessel motion and related processes. Fractals 9, 329–346.Google Scholar
Baeumer, B. and Meerschaert, M. M. (2001). Stochastic solutions for fractional Cauchy problems. Fractional Calculus Appl. Anal. 4, 481–500.Google Scholar
Barkai, E., Metzler, R. and Klafter, J. (2000). From continuous time random walks to the fractional Fokker–Planck equation. Phys. Rev. E 61, 132–138.CrossRefGoogle Scholar
Benson, D. A., Wheatcraft, S. W. and Meerschaert, M. M. (2000). The fractional-order governing equation of Lévy motion. Water Resour. Res. 36, 1413–1424.Google Scholar
Bonaccorsi, S. and Tubaro, L. (2003). Mittag-Leffler's function and stochastic linear Volterra equations of convolution type. Stoch. Anal. Appl. 21, 61–78.Google Scholar
Cairoli, R. and Walsh, J. B. (1975). Stochastic integrals in the plane. Acta Math. 134, 111–181.Google Scholar
Caputo, M. (1967). Linear model of dissipation whose Q is almost frequency independent. II. Geophys. J. R. Astr. Soc. 13, 529–539.CrossRefGoogle Scholar
Chaves, A. (1998). A fractional diffusion equation to describe Lévy flights. Phys. Lett. A 239, 13–16.Google Scholar
Compte, A. and Metzler, R. (1997). The generalised Cattaneo equation for the description of anomalous transport processes. J. Phys. A 30, 7277–7289.CrossRefGoogle Scholar
Davies, E. B. (1995). Spectral Theory and Differential Operators (Camb. Studies Adv. Math. 42). Cambridge University Press.Google Scholar
Djrbashian, M. M. (1993). Harmonic Analysis and Boundary Value Problems in the Complex Domain (Operator Theory Adv. Appl. 65). Birkhäuser, Basel.Google Scholar
Djrbashian, M. M. and Nersesian, A. B. (1968). Fractional derivatives and the Cauchy problem for differential equations of fractional order. Izv. Akad. Nauk Armjanskvy SSR 3, 3–29 (in Russian).Google Scholar
Friedlander, F. G. and Joshi, M. S. (1998). Introduction to the Theory of Distributions, 2nd edn. Cambridge University Press.Google Scholar
Gorenflo, R., Iskenderov, A. and Luchko, Y. (2000). Mapping between solutions of fractional diffusion–wave equations. Fractional Calculus Appl. Anal. 3, 75–86.Google Scholar
Gorenflo, R., Luchko, Y. and Mainardi, F. (1999). Analytical properties and applications of the Wright function. Fractional Calculus Appl. Anal. 2, 383–414.Google Scholar
Gorenflo, R., Luchko, Y. and Mainardi, F. (2000). Wright functions and scale-invariate solutions of the diffusion–wave equation. J. Comput. Appl. Math. 118, 175–191.Google Scholar
Gorenflo, R., Mainardi, F., Scalas, E. and Raberto, M. (2001). Fractional calculus and continuous-time finance. III. The diffusion limit. In Mathematical Finance, eds Kohlmann, M. and Tang, S., Birkhäuser, Basel, pp. 171–180.Google Scholar
Hilfer, R. (2000). Fractional time evolution. In Applications of Fractional Calculus in Physics, ed. Hilfer, R., World Scientific, Singapore, pp. 87–130.CrossRefGoogle Scholar
Hughes, B. D. (1995). Random Walks and Random Environments, Vol. 1. Oxford University Press.Google Scholar
Jiménez, J. (1994). Hyperviscous vortices. J. Fluid Mech. 279, 169–176.Google Scholar
Kallenberg, O. (2002). Foundations of Modern Probability. Springer, New York.Google Scholar
Klafter, J., Blumen, A. and Shlesinger, M. F. (1987). Stochastic pathways to anomalous diffusion. Phys. Rev. A 35, 3081–3085.Google Scholar
Kochubei, A. N. (1990). Fractional order diffusion. J. Differential Equat. 26, 485–492.Google Scholar
Lasota, A. and Mackey, M. C. (1995). Chaos, Fractals, and Noise (Appl. Math. Sci. 97), 2nd edn. Springer, New York.Google Scholar
Leveque, E. and She, Z.-S. (1995). Viscous effects on inertial range scalings in a dynamical model of turbulence. Phys. Rev. Lett. 75, 2690–2693.Google Scholar
Lukacs, E. (1970). Characteristic Functions. Griffin, London.Google Scholar
Mainardi, F. (1996). The fundamental solutions for the fractional diffusion–wave equation. Appl. Math. Lett. 9, 23–28.Google Scholar
Mainardi, F., Raberto, M., Gorenflo, R. and Scalas, E. (2000). Fractional calculus and continuous-time finance. II. The waiting-time distribution. Physica A 287, 468–481.Google Scholar
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
Meerschaert, M. M., Benson, D. A. and Baeumer, B. (1999). Multidimensional advection and fractional dispersion. Phys. Rev. E 59, 5026–5028.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
Metzler, R. and Klafter, J. (2000). The random walk's guide to anomalous diffusion: a fractional dynamic approach. Phys. Rep. 339, 1–77.CrossRefGoogle Scholar
Orsingher, E. and Beghin, L. (2004). Time-fractional telegraph equations and telegraph processes with Brownian time. Prob. Theory Relat. Fields 128, 141–160.CrossRefGoogle Scholar
Podlubny, I. (1999). Fractional Differential Equations. Academic Press, San Diego, CA.Google Scholar
Rajput, B. and Rosiński, J. (1989). Spectral representation of infinitely divisible processes. Prob. Theory Relat. Fields 82, 451–487.Google Scholar
Rozanov, Y. A. (1998). Random Fields and Stochastic Partial Differential Equations. Kluwer, Boston, MA.Google Scholar
Saichev, A. I. and Zaslawsky, G. M. (1997). Fractional kinetic equations: solutions and applications. Chaos 7, 753–764.Google Scholar
Schneider, W. R. (1990). Fractional diffusion. In Dynamics and Stochastic Processes, Theory and Applications (Lecture Notes Phys. 355), eds Lima, R., Streit, L. and Mendes, D. V., Springer, Heidelberg, pp. 276–286.Google Scholar
Schneider, W. R. and Wyss, W. (1989). Fractional diffusion and wave equations. J. Math. Phys. 30, 134–144.Google Scholar
Shlesinger, M., Klafter, J. and Wong, Y. M. (1982). Random walks with infinite spatial and temporal moments. J. Statist. Phys. 27, 499–512.Google Scholar
Uchaikin, V. V. and Zolotarev, V. M. (1999). Chance and Stability. Stable Distributions and Their Applications. VSP, Utrecht.Google Scholar
Walsh, J. B. (1981). A stochastic model of neural response. Adv. Appl. Prob. 13, 231–281.Google Scholar
Walsh, J. B. (1984). Regularity properties of a stochastic partial differential equation. In Seminar on Stochastic Processes (Gainesville, FL, 1983; Progress Prob. Statist. 7), Birkhäuser, Boston, MA, pp. 257–290.Google Scholar
Wong, E. and Hajek, B. (1985). Stochastic Processes in Engineering Systems. Springer, New York.Google Scholar