Renewal processes of Mittag-Leffler and Wright type

After sketching the basic principles of renewal theory we discuss the classical Poisson process and offer two other processes, namely the renewal process of Mittag-Leffler type and the renewal process of Wright type, so named by us because special functions of Mittag-Leffler and of Wright type appear in the definition of the relevant waiting times. We compare these three processes with each other, furthermore consider corresponding renewal processes with reward and numerically their long-time behaviour.

[1]  F. Mainardi,et al.  The fundamental solution of the space-time fractional diffusion equation , 2007, cond-mat/0702419.

[2]  Sheldon M. Ross,et al.  Introduction to Probability Models (4th ed.). , 1990 .

[3]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[4]  Enrico Scalas,et al.  Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[6]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance II: the waiting-time distribution , 2000, cond-mat/0006454.

[7]  Sheldon M. Ross,et al.  Introduction to probability models , 1975 .

[8]  E. Barkai.  CTRW pathways to the fractional diffusion equation , 2001, cond-mat/0108024.

[9]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance , 2000, cond-mat/0001120.

[10]  Francesco Mainardi,et al.  Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics , 2012, 1201.0863.

[11]  Francesco Mainardi,et al.  Random walk models approximating symmetric space-fractional diffusion processes , 2012, 1210.6589.

[12]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[13]  Francesco Mainardi,et al.  The Wright functions as solutions of the time-fractional diffusion equation , 2003, Appl. Math. Comput..

[14]  Francesco Mainardi,et al.  On Mittag-Leffler-type functions in fractional evolution processes , 2000 .

[15]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[16]  R. Gorenflo,et al.  Analytical properties and applications of the Wright function , 2007, math-ph/0701069.

[17]  I. M. Sokolov,et al.  Distributed-Order Fractional Kinetics , 2004 .

[18]  Mark M. Meerschaert,et al.  STOCHASTIC SOLUTIONS FOR FRACTIONAL CAUCHY PROBLEMS , 2003 .

[19]  Rudolf Hilfer,et al.  On fractional diffusion and continuous time random walks , 2003 .

[20]  Elliott W. Montroll,et al.  Random walks on lattices. IV. Continuous-time walks and influence of absorbing boundaries , 1973 .

[21]  Evolution, its Fractional Extension and Generalization , 1999, math-ph/9912023.

[22]  Francesco Mainardi,et al.  Simply and multiply scaled diffusion limits for continuous time random walks , 2005 .

[23]  E. Montroll,et al.  Random Walks on Lattices. II , 1965 .

[24]  Hilfer,et al.  Fractional master equations and fractal time random walks. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[25]  Francesco Mainardi,et al.  Mellin transform and subordination laws in fractional diffusion processes , 2007, math/0702133.

[26]  Francesco Mainardi,et al.  Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk , 2007, 0709.3990.

[27]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[28]  Hans-Peter Scheffler,et al.  Stochastic solution of space-time fractional diffusion equations. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  Rudolf Gorenflo,et al.  From Power Laws to Fractional Diffusion: the Direct Way , 2007, 0801.0142.

[30]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[31]  G. Weiss.  Aspects and Applications of the Random Walk , 1994 .

[32]  Linear response in complex systems: CTRW and the fractional Fokker-Planck equations , 2001, cond-mat/0107632.

[33]  E. Bazhlekova.  Subordination principle for fractional evolution equations , 1999 .

[34]  Barkai,et al.  From continuous time random walks to the fractional fokker-planck equation , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[35]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[36]  Enrico Scalas,et al.  Fractional Calculus and Continuous-Time Finance III : the Diffusion Limit , 2001 .

[37]  Francesco Mainardi,et al.  The time fractional diffusion-wave equation , 1995 .

[38]  F. Mainardi.  Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena , 1996 .

[39]  R. Gorenflo,et al.  A fractional generalization of the Poisson processes , 2007, math/0701454.

[40]  J. Corcoran.  Modelling Extremal Events for Insurance and Finance , 2002 .

[41]  E. Montroll.  Random walks on lattices , 1969 .

[42]  Alexander I. Saichev,et al.  Fractional kinetic equations: solutions and applications. , 1997, Chaos.

[43]  I. Podlubny.  Fractional differential equations , 1998 .

[44]  W. Schneider,et al.  Fractional diffusion and wave equations , 1989 .

[45]  R. Gorenflo,et al.  Wright functions as scale-invariant solutions of the diffusion-wave equation , 2000 .

[46]  V. Kiryakova.  Generalized Fractional Calculus and Applications , 1993 .