# 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)
Some recent advances in theory and simulation of fractional diffusion processes. (English) Zbl 1166.45004
Summary: To offer an insight into the rapidly developing theory of fractional diffusion processes, we describe in some detail three topics of current interest: (i) the well-scaled passage to the limit from continuous time random walk under power law assumptions to space-time fractional diffusion, (ii) the asymptotic universality of the Mittag-Leffler waiting time law in time-fractional processes, (iii) our method of parametric subordination for generating particle trajectories.

##### MSC:
 45K05 Integro-partial differential equations 26A33 Fractional derivatives and integrals (real functions) 60G18 Self-similar processes 60G50 Sums of independent random variables; random walks 60G51 Processes with independent increments; Lévy processes 60J60 Diffusion processes
Full Text:
##### References:
 [1] Andries, E.; Umarov, S.; Steinberg, S.: Monte Carlo random walk simulation based on distributed order differential equations with applications to cell biology, Fractional calculus and applied analysis 9, 351-370 (2006) · Zbl 1132.65114 [2] Baeumer, B.; Meerschaert, M. M.: Stochastic solutions for fractional Cauchy problems, Fractional calculus and applied analysis 4, 481-500 (2001) · Zbl 1057.35102 [3] Barkai, E.: Fractional Fokker-Planck equation, solution, and application, Physical review E 63 (2001) [4] Bingham, N. H.; Goldie, C. M.; Teugels, J. L.: Regular variation, (1987) [5] Chechkin, A. V.; Gorenflo, R.; Sokolov, I. M.: Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Physical review E 66, 046129/1-6 (2002) [6] Chechkin, A. V.; Gorenflo, R.; Sokolov, I. M.; Gonchar, V. Yu.: Distributed order time fractional diffusion equation, Fractional calculus and applied analysis 6, 259-279 (2002) · Zbl 1089.60046 [7] Cox, D. R.: Renewal theory, (1967) · Zbl 0168.16106 [8] Feller, W.: An introduction to probability theory and its applications, An introduction to probability theory and its applications 2 (1971) · Zbl 0219.60003 [9] Fulger, D.; Scalas, E.; Germano, G.: Monte-Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation, Physical review E 77 (2008) [10] Gel?fand, I. M.; Shilov, G. E.: Generalized functions, Generalized functions (1964) · Zbl 0115.33101 [11] Gnedenko, B. V.; Kolmogorov, A. N.: Limit distributions for sums of independent random variables, (1954) · Zbl 0056.36001 [12] B.V. Gnedenko, I.N. Kovalenko, Introduction to Queueing Theory, Israel Program for Scientific Translations, Jerusalem (1968) · Zbl 0186.24502 [13] Gorenflo, R.; Abdel-Rehim, E.: From power laws to fractional diffusion: the direct way, Vietnam journal of mathematics 32, No. SI, 65-75 (2004) · Zbl 1086.60049 [14] Gorenflo, R.; Mainardi, F.: Fractional calculus: integral and differential equations of fractional order, Fractals and fractional calculus in continuum mechanics, 223-276 (1997) [15] Gorenflo, R.; Mainardi, F.: Random walk models for space-fractional diffusion processes, Fractional calculus and applied analysis 1, 167-191 (1998) · Zbl 0946.60039 [16] R. Gorenflo, F. Mainardi, Non-Markovian random walks, scaling and diffusion limits, in: O.E. Barndorff-Nielsen (Eds.), Mini-Proceedings:e 2-nd MaPhySto Conference on Lévy Processes: Theory and Applications, Dept. Mathematics, University of Aarhus, Denmark, 21--25 January 2002, [Available c/o http://www.maphysto.dk, Miscellanea no. 22] [17] Gorenflo, R.; Mainardi, F.: Simply and multiply scaled diffusion limits for continuous time random walks, Journal of physics: conference series 7, 1-16 (2005) [18] R. Gorenflo, F. Mainardi, Continuous time random walk, Mittag--Leffler waiting time and fractional diffusion: mathematical aspects, in: R. Klages, G. Radons and I.M. Sokolov (Eds.), Anomalous Transport: Foundations and Applications, Wiley-VCH, Weinheim, Germany (2008) (in press) [E-print http://arxiv.org/abs/0705.0797] [19] 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 [20] Gorenflo, R.; Mainardi, F.; Vivoli, A.: Continuous time random walk and parametric subordination in fractional diffusion, Chaos, solitons and fractals 34, 87-103 (2007) · Zbl 1142.82363 · doi:10.1016/j.chaos.2007.01.052 [21] Hilfer, R.; Anton, L.: Fractional master equations and fractal time random walks, Physical review E 51, R848-R851 (1995) [22] Ilic, M.; Liu, F.; Turner, I.; Anh, V.: Numerical approximation of a fractional-in-space diffusion equation (II) with non-homogeneous boundary conditions, Fractional calculus and applied analysis 9, 333-349 (2006) · Zbl 1132.35507 [23] 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 (2001) [24] A. Janicki, Numerical and Statistical Approximation of Stochastic Differential Equations with Non-Gaussian Measures, Monograph No 1, H. Steinhaus Center for Stochastic Methods in Science and Technology, Technical University, Wroclaw, Poland 1996 [25] Janicki, A.; Weron, A.: Simulation and chaotic behavior of ${\alpha}$--stable stochastic processes, (1994) · Zbl 0955.60508 [26] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006) · Zbl 1092.45003 [27] Kleinhans, D.; Friedrich, R.: Continuous time random walks: simulation of continuous trajectories, Physical review E 76 (2008) [28] Mainardi, F.; Luchko, Yu.; Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation, Fractional calculus and applied analysis 4, 153-192 (2001) · Zbl 1054.35156 [29] Mainardi, F.; Pagnini, G.; Saxena, R. K.: Fox H functions in fractional diffusion, Journal of computational and applied mathematics 178, 321-331 (2005) · Zbl 1061.33012 · doi:10.1016/j.cam.2004.08.006 [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] Meerschaert, M. M.; Benson, D. A.; Scheffler, H. -P.; Baeumer, B.: Stochastic solutions of space-fractional diffusion equation, Physical review E 65, 041103-1/4 (2002) · Zbl 1244.60080 [32] Meerschaert, M. M.; Benson, D. A.; Scheffler, H. -P.; Becker-Kern, P.: Governing equations and solutions of anomalous random walk limits, Physical review E 66, 060102-1/4 (2002) [33] Meerschaert, M. M.; Scheffler, H. -P.: Limit distributions for sums of independent random vectors: heavy tails in theory and practice, (2001) · Zbl 0990.60003 [34] Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Physical reports 339, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3 [35] Montroll, E. W.; Scher, H.: Random walks on lattices, IV: Continuous-time walks and influence of absorbing boundaries, Journal of statistical physics 9, 101-135 (1973) [36] Montroll, E. W.; Weiss, G. H.: Random walks on lattices, II, Journal of mathematical physics 6, 167-181 (1965) [37] Newman, M. E. J.: Power laws, Pareto distributions and Zipf’s law, Contemporary physics 46, 323-351 (2005) [38] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008 [39] Piryatinska, A.; Saichev, A. I.; Woyczynski, W. A.: Models of anomalous diffusion: the subdiffusive case, Physica A 349, 375-420 (2005) [40] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003 [41] Sato, K-I.: Lévy processes and infinitely divisible distributions, (1999) [42] Scalas, E.: The application of continuous-time random walks in finance and economics, Physica A 362, 225-239 (2006) [43] Scalas, E.; Gorenflo, R.; Mainardi, F.: Fractional calculus and continuous-time finance, Physica A 284, 376-384 (2000) [44] Scalas, E.; Gorenflo, R.; Mainardi, F.: Uncoupled continuous-time random walks: solution and limiting behavior of the master equation, Physical review E 69, 011107-1/8 (2004) [45] Schroeder, M.: Fractals, chaos, power laws, (1991) · Zbl 0758.58001 [46] Sokolov, I. M.; Chechkin, A. V.; Klafter, J.: Distributed-order fractional kinetics, Acta physica polonica 35, 1323-1341 (2004) [47] 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 · doi:10.1063/1.1860472 [48] Weiss, G. H.: Aspects and applications of random walks, (1994) · Zbl 0925.60079