zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Continuous-time random walk and parametric subordination in fractional diffusion. (English) Zbl 1142.82363
Summary: The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW) is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. We treat the CTRW as a combination of a random walk on the axis of physical time with a random walk in space, both walks happening in discrete operational time. In the continuum limit, we obtain a (generally non-Markovian) diffusion process governed by a space-time fractional diffusion equation. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. By what we call parametric subordination, applied to a combination of a Markov process with a positively oriented Lévy process, we generate and display sample paths for some special cases.

MSC:
82C41Dynamics of random walks, random surfaces, lattice animals, etc.
82C70Transport processes (time-dependent statistical mechanics)
WorldCat.org
Full Text: DOI arXiv
References:
[1] Baeumer, B.; Meerschaert, M. M.: Stochastic solutions for fractional Cauchy problems. Fract calcul appl anal 4, 481-500 (2001) · Zbl 1057.35102
[2] Balescu, R.: Statistical dynamics: matter out of equilibrium. (1994) · Zbl 0997.82505
[3] Barkai, E.: Fractional Fokker -- Planck equation, solution, and application. Phys rev E 63 (2001)
[4] Barkai, E.: CTRW pathways to the fractional diffusion equation. Chem phys 284, 13-27 (2002)
[5] Barkai, E.; Metzler, R.; Klafter, J.: From continuous time random walk to fractional Fokker -- Planck equation. Phys rev E 61, 132-138 (2000)
[6] Barndorff-Nielsen, O. E.; Mikosch, T.; Resnick, S. I.: Lévy processes: theory and applications. (2001) · Zbl 0961.00012
[7] Bochner, S.: Harmonic analysis and the theory of probability. (1955) · Zbl 0068.11702
[8] Bochner, S.: Subordination of non-Gaussian stochastic processes. Proc natl acad sci USA 48, 19-22 (1962) · Zbl 0105.33002
[9] Cox, D. R.: Renewal theory. (1967) · Zbl 0168.16106
[10] Feller, W.: An introduction to probability theory and its applications. 2 (1971) · Zbl 0219.60003
[11] Gel&grave, I. M.; Fand; Shilov, G. E.: Generalized functions. (1964)
[12] Gorenflo, R.; Abdel-Rehim, E.: From power laws to fractional diffusion: the direct way. Viet J math 32, No. SI, 65-75 (2004) · Zbl 1086.60049
[13] Gorenflo, R.; Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. Fractals and fractional calculus in continuum mechanics, 223-276 (1997)
[14] Gorenflo, R.; Mainardi, F.: Fractional diffusion processes: probability distributions and continuous time random walk. Lecture notes in physics 621, 148-166 (2003)
[15] Gorenflo R, Mainardi F, Simply and multiply scaled diffusion limits for continuous time random walks. In: Benkadda S, Leoncini X, Zaslavsky G. editors. Proceedings of the international workshop on chaotic transport and complexity in fluids and plasmas Carry Le Rouet (France) 20 -- 25 June 2004, IOP (Institute of Physics) Journal of Physics: Conference Series 7; 2005. p. 1 -- 16.
[16] Gorenflo, R.; Mainardi, F.; Scalas, E.; Raberto, M.: Fractional calculus and continuous-time finance III: The diffusion limit. Mathematical finance, 171-180 (2001) · Zbl 1138.91444
[17] Grigolini, P.; Rocco, A.; West, B. J.: Fractional calculus as a macroscopic manifestation of randomness. Phys rev E 59, 2603-2613 (1999)
[18] Hilfer, R.: Exact solutions for a class of fractal time random walks. Fractals 3, 211-216 (1995) · Zbl 0881.60066
[19] Hilfer, R.: Applications of fractional calculus in physics. (2000) · Zbl 0998.26002
[20] Hilfer, R.: On fractional diffusion and continuous time random walks. Physica A 329, 35-39 (2003) · Zbl 1029.60033
[21] Hilfer, R.; Anton, L.: Fractional master equations and fractal time random walks. Phys rev E 51, R848-R851 (1995)
[22] Jacob N. Pseudodifferential operators -- Markov processes, vol. I: Fourier analysis and semigroups vol. II: Generators and their potential theory, vol. III: Markov Processes and Applications, Imperial College Press, London (2001), (2002), (2005).
[23] Janicki, A.: Numerical and statistical approximation of stochastic differential equations with non-Gaussian measures monograph, no. 1. (1996)
[24] Janicki, A.; Weron, A.: Simulation and chaotic behavior of $\alpha $-stable stochastic processes. (1994) · Zbl 0955.60508
[25] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations. (2006) · Zbl 1092.45003
[26] Kotulski, M.: Asymptotic distributions of continuous-time random walks: a probabilistic approach. J stat phys 81, 777-792 (1995) · Zbl 1107.60318
[27] Mainardi, F.; Luchko, Yu.; Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. Fract calcul appl anal 4, 153-192 (2001) · Zbl 1054.35156
[28] Mainardi, F.; Pagnini, G.; Gorenflo, R.: Mellin transform and subordination laws in fractional diffusion processes. Fract calcul appl anal 6, 441-459 (2003) · Zbl 1083.60032
[29] Mainardi, F.; Pagnini, G.; Saxena, R. K.: Fox H functions in fractional diffusion. J computat appl math 178, 321-331 (2005) · Zbl 1061.33012
[30] Mainardi, F.; Raberto, M.; Gorenflo, R.; Scalas, E.: Fractional calculus and continuous-time finance II: The waiting-time distribution. Physica A 287, 468-481 (2000) · Zbl 1138.91444
[31] Mannella, R.; Grigolini, P.; West, B. J.: A dynamical approach to fractional Brownian motion. Fractals 2, 81-94 (1994)
[32] Meerschaert, M. M.; Benson, D. A.; Scheffler, H. P.; Baeumer, B.: Stochastic solutions of space-fractional diffusion equation. Phys rev E 65 (2002) · Zbl 1244.60080
[33] Meerschaert, M. M.; Benson, D. A.; Scheffler, H. P.; Becker-Kern, P.: Governing equations and solutions of anomalous random walk limits. Phys rev E 66 (2002)
[34] Metzler, R.; Klafter, J.; Sokolov, I. M.: Anomalous transport in external fields: continuous time random walks and fractional diffusion equations extended. Phys rev E 58, 1621-1633 (1998)
[35] Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys rep 339, 1-77 (2000) · Zbl 0984.82032
[36] 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, R161-R208 (2004) · Zbl 1075.82018
[37] Montroll, E. W.; Scher, H.: Random walks on lattices IV: Continuous-time walks and influence of absorbing boundaries. J stat phys 9, 101-135 (1973)
[38] Montroll, E. W.; Shlesinger, M. F.: On the wonderful world of random walks. Nonequilibrium phenomena II: From stochastics to hydrodynamics, 1-121 (1984)
[39] Montroll, E. W.; Weiss, G. H.: Random walks on lattices II. J math phys 6, 167-181 (1965)
[40] Montroll, E. W.; West, B. J.: On an enriched collection of stochastic processes. Fluctuation phenomena, 61-175 (1979)
[41] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008
[42] Piryatinska, A.; Saichev, A. I.; Woyczynski, W. A.: Models of anomalous diffusion: the subdiffusive case. Physica A 349, 375-420 (2005)
[43] Saichev, A.; Zaslavsky, G.: Fractional kinetic equations: solutions and applications. Chaos 7, 753-764 (1997) · Zbl 0933.37029
[44] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications. (1993) · Zbl 0818.26003
[45] Sato, K-I.: Lévy processes and infinitely divisible distributions. (1999)
[46] Scalas, E.: The application of continuous-time random walks in finance and economics. Physica A 362, 225-239 (2006)
[47] Scalas, E.; Gorenflo, R.; Mainardi, F.: Fractional calculus and continuous-time finance. Physica A 284, 376-384 (2000)
[48] Scalas, E.; Gorenflo, R.; Mainardi, F.: Uncoupled continuous-time random walks: solution and limiting behavior of the master equation. Phys rev E 69 (2004) · Zbl 1087.60064
[49] Shlesinger, M. F.; Zaslavsky, G. M.; Klafter, J.: Strange kinetics. Nature 363, 31-37 (1993)
[50] Sokolov, I. M.: Lévy flights from a continuous-time process. Phys rev E 63 (2001)
[51] Sokolov, I. M.: Thermodynamics and fractional Fokker -- Planck equation. Phys. rev. E 63 (2001)
[52] Sokolov, I. M.: Solutions of a class of non-Markovian Fokker -- Planck equations. Phys rev E 66 (2002)
[53] Sokolov, I. M.; Klafter, J.; Blumen, A.: Linear response in complex systems: CTRW and the fractional Fokker -- Planck equations. Physica A 302, 268-278 (2001) · Zbl 0983.60040
[54] Sokolov, I. M.; Klafter, J.: From diffusion to anomalous diffusion: a century after Einstein’s Brownian motion. Chaos 15, 026103-026109 (2005) · Zbl 1080.82022
[55] Sokolov, I. M.; Klafter, J.; Blumen, A.: Fractional kinetics. Phys today 55, 48-54 (2002)
[56] Stanislavski, A. A.: Memory effects and macroscopic manifestation of randomness. Phys rev E 61, 4752-4759 (2000)
[57] Stanislavsky, A. A.: Black -- Scholes model under subordination. Physica A 318, 469-474 (2003) · Zbl 1010.91029
[58] Uchaikin, V. V.; Saenko, V. V.: Stochastic solution of partial differential equations of fractional orders. Siber J numer math 6, 197-203 (2003) · Zbl 1032.60057
[59] Uchaikin, V. V.; Zolotarev, V. M.: Chance and stability stable distributions and their applications. (1999) · Zbl 0944.60006
[60] Weiss, G. H.: Aspects and applications of random walks. (1994) · Zbl 0925.60079
[61] West, B. J.; Bologna, M.; Grigolini, P.: Physics of fractal operators. (2003)
[62] Wyss, M. M.; Wyss, W.: Evolution, its fractional extension and generalization. Fract calcul appl anal 4, 273-284 (2001) · Zbl 1042.45005
[63] Zaslavsky, G. M.: Chaos, fractional kinetics and anomalous transport. Phys rep 371, 461-580 (2002) · Zbl 0999.82053
[64] Zaslavsky, G. M.: Hamiltonian chaos and fractional dynamics. (2005) · Zbl 1083.37002