January 2001 - 010103
Fractional diffusion processes: recent papers by "fracalmo" promoters

The are so many papers on fractional diffusion processes published in several journals (mathematics, physics, engineering, finance,...) that any list of references on the subject is far to be exhaustive. Furthermore, the number of papers is growing so quickly that an up-date bibliography is (almost) impossible. This interest is increasing in spite of several people who do not trust in the physical and/or mathematical relevance of fractional calculus!

Here we would like to present a preliminary list of recent references related to our approach to fractional diffusion, based upon the Caputo fractional derivative for time and the Riesz-Feller fractional derivative for space. The authors of these papers are thus the promoters of this WEB site and their associates. We shall refer to papers of other authors in future NEWS; in a next news we shall refer to recent papers by Professor M. Caputo.

We divide our papers (some of which already downloadable, others soon) in 3 groups:
1) Generalities and solutions for fractional diffusion equations
2) Random walk models for fractional diffusion
3) Applications to continuous-time finance

1) Generalities and solutions for fractional diffusion equations

P. Paradisi, R. Cesari, F. Mainardi and F. Tampieri (2001):
The fractional Fick's law for non-local transport processes,
Physica A, Vol. 293, No 1-2, 130-142 (2001)

F. Mainardi, Yu Luchko and G. Pagnini (2001):
The fundamental solutions of the space-time fractional diffusion equation,
Fractional Calculus and Applied Analysis, Vol. 4, No 2 (2001), pp 153-192.
[Paper dedicated to Professor Rudolf Gorenflo for his 70-th birthday]
[preprint downloadable]

R. Gorenflo, Yu. Luchko and F. Mainardi (2000):
Wright functions as scale-invariant solutions of the diffusion-wave equation,
J. Computational and Appl. Mathematics, Vol. 118 No 1-2, 175-191 (2000).

F. Mainardi and F. Tampieri (1999):
Diffusion regimes in Brownian motion induced by the Basset history force,
Technical Report No 1 (ISAO-TP-99/1), ISAO-CNR, Bologna, March 1999, pp. 24.
[Invited Lecture, Meeting of the TAO Working Group on Diffusion, Stockholm, Sweden, 6-11 October 1997]

R. Gorenflo and F. Mainardi (1999):
Feller fractional diffusion and Levy stable motions,
Mini Proceedings of the International Conference on "Levy Processes: Theory and Applications", MaPhySto (Mathematical Physics and Stochastics Centre), Department of Mathematics, University of Aarhus, Denmark 18-22 January 1999. MPS-misc. 1999-11, April 1999, pp. 11-122, edited by O.E. Barndorff-Nielsen, S.E. Graversen and T. Mikosch.
[preprint downloadable]

R. Gorenflo and F. Mainardi (1998):
Signalling problem and Dirichlet-Neumann map for time-fractional diffusion-wave equations,
Matimyas Matematika, Vol. 21, August 1998, pp. 109-118.
[Special issue of Matimyas Matematika (Official Journal of the Mathematical Society of the Philippines) edited by R. Gorenflo and M. P. Navarro, Proceedings of the International Conference on "Inverse Problems and Applications", University of the Philippines - Diliman, February 23-27, 1998, ISSN 0115-6926]
Preprint No. A-07/98, Freie Universitat Berlin, Serie A Mathematik (1998). [http://www.math.fu-berlin.de/publ]

R. Gorenflo and F. Mainardi :
Fractional calculus and stable probability distributions,
Archives of Mechanics, Vol. 50 (3), 377-388 (1998).
[preprint downloadable]

F. Mainardi, P. Paradisi and R. Gorenflo (1997):
Probability distributions generated by fractional diffusion equations, to appear in J. Kertesz and I. Kondor (Editors), "Econophysics: an Emerging Science", Kluwer, Dordrecht ????, pp. 39.
[Invited Lecture, Workshop on Econophysics, Budapest 21-27 July 1997]
[preprint downloadable]

F. Mainardi and M. Tomirotti (1997):
Seismic pulse propagation with constant Q and stable probability distributions,
Annali Geofisica, Vol. 40, 1311-1328 (1997).
[Paper dedicated to Professor Michele CAPUTO for his 70-th birthday]

F. Mainardi (1997):
Fractional Calculus, Some Basic Problems in Continuum and Statistical Mechanics,
in A. Carpinteri and F. Mainardi (Editors), "Fractals and Fractional Calculus in Continuum Mechanics", Springer Verlag, Wien and New York (1997), pp. 291-348.
[Vol. no 378, series CISM Courses and Lecture Notes, ISBN 3-211-82913-X]
[Lecture Notes of the Advanced School held at CISM, Udine, Italy, 23-27 September 1996]

2) Random walk models for fractional diffusion

R. Gorenflo, P. Paradisi, F. Mainardi and D. Moretti (2001)
Random walk models for space-time fractional diffusion,
Pre-print, to appear.

R. Gorenflo and F. Mainardi (2000):
Random walk models approximating symmetric space-fractional diffusion processes,
in: J. Elschner, I. Gohberg and B. Silbermann (Editors), Problems in Mathematical Physics (Siegfried Prossdorf Memorial Volume). Proceedings of the 11-th TMP Conference. (Birkhauser Verlag, Boston-Basel-Berlin, 2000), pp. 120-145

R. Gorenflo and F. Mainardi (2000):
Approximation of Levy-Feller diffusion by random walk,
J. Analysis and its Applications (ZAA), Vol. 18 (2), 231-246 (1999).
[preprint downloadable]

R. Gorenflo, G. De Fabritiis and F. Mainardi (1999):
Discrete random walk models for symmetric Levy-Feller diffusion processes,
Physica A, Vol. 269 (1) 79--89 (1999).
[preprint downloadable or at http://xxx.lanl.gov/abs/cond-mat/9903264]
[Special issue of Physica A, edited by R.N. Mantegna, containing the Proceedings of the International Workshop on "Econophysics and Statistical Finance", Palermo (Italy), 28-30 September 1998]

R. Gorenflo and F. Mainardi (1998):
Random walk models for space-fractional diffusion processes,
Fractional Calculus and Applied Analysis, Vol. 1 (2), 167-191 (1998).
[preprint downloadable]

3) Applications in Continuous-Time Finance

R. Gorenflo, F. Mainardi, M. Raberto and E. Scalas :
Fractional diffusion in finance: basic theory,
in G.I. Bischi and L. Gardini (Editors), Proceedings of the Workshop Modelli Dinamici in Economia e Finanza (MDEF2000), Urbino, September 28-30, 2000. [www.econ.uniurb.it/bischi/mdef2000.htm]

R. Gorenflo, F. Mainardi, E. Scalas and M. Raberto (2000):
Fractional calculus and continuous-time finance III: the diffusion limit,
Pre-print, to appear.

F. Mainardi, M. Raberto, R. Gorenflo and E. Scalas (2000):
Fractional calculus and continuous-time finance II: the waiting-time distribution,
Physica A, Vol. 287, No 3-4, 468--481 (2000).
[http://xxx.lanl.gov/abs/cond-mat/0006454]
[Special issue of Physica A, edited by F. Schweitzer and D. Helbing, containing the Proceedings of the International Workshop on "Economic Dynamics from the Physics Point of View", Bad-Honnef (Germany), 27-30 March 2000]

E. Scalas, R. Gorenflo and F. Mainardi :
Fractional calculus and continuous-time finance,
Physica A, Vol. 284, No 1-4, 376-384 (2000)
[Preprint downloadable from http://xxx.lanl.gov/abs/cond-mat/0001120]