Hostname: page-component-669899f699-cf6xr Total loading time: 0 Render date: 2025-04-28T17:13:12.666Z Has data issue: false hasContentIssue false
  • English
  • Français

Fractional Relaxation Equations and Brownian Crossing Probabilities of a Random Boundary

Part of: Other special functions Stochastic processes General theory in ordinary differential equations

Published online by Cambridge University Press:  04 January 2016

L. Beghin
L. Beghin*
Affiliation:
Sapienza University of Rome
*
∗ Postal address: Department of Statistical Sciences, Sapienza University of Rome, Piazzale Aldo Moro 5, 00185 Rome, Italy. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we analyze different forms of fractional relaxation equations of order ν ∈ (0, 1), and we derive their solutions in both analytical and probabilistic forms. In particular, we show that these solutions can be expressed as random boundary crossing probabilities of various types of stochastic process, which are all related to the Brownian motion B. In the special case ν = ½, the fractional relaxation is shown to coincide with Pr{sup0≤s≤tB(s) < U} for an exponential boundary U. When we generalize the distributions of the random boundary, passing from the exponential to the gamma density, we obtain more and more complicated fractional equations.

Keywords

Fractional relaxation equationgeneralized Mittag-Leffler functionprocesses with random timereflecting and elastic Brownian motioniterated Brownian motionboundary crossing probability

MSC classification

Primary: 60G15: Gaussian processes 34A08: Fractional differential equations 33E12: Mittag-Leffler functions and generalizations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 2 , June 2012 , pp. 479 - 505
Copyright
© Applied Probability Trust 

References

Beghin, L. (2012). Random-time processes governed by differential equations of fractional distributed order. To appear in Chaos Solitons Fractals.Google Scholar
Beghin, L. and Orsingher, E. (2009). Fractional Poisson processes and related planar random motions. Electron. J. Prob. 14, 1790–1827.Google Scholar
Beghin, L. and Orsingher, E. (2009). Iterated elastic Brownian motions and fractional diffusion equations. Stoch. Process. Appl. 119, 1975–1975.Google Scholar
Beghin, L. and Orsingher, E. (2010). Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron. J. Prob. 15, 684–709.Google Scholar
Beghin, L. and Orsingher, E. (2012). Poisson process with different Brownian clocks. Stochastics 84, 79–112.Google Scholar
Beghin, L., Orsingher, E. and Ragozina, T. (2001). Joint distributions of the maximum and the process for higher-order diffusions. Stoch. Process. Appl. 94, 71–93.Google Scholar
Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121). Cambridge University Press.Google Scholar
Chechkin, A. V., Gorenflo, R. and Sokolov, I. M. (2002). Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E 66, 046129, 7 pp.Google Scholar
D'Ovidio, M. and Orsingher, E. (2011). Bessel processes and hyperbolic Brownian motions stopped at different random times. Stoch. Process. Appl. 121, 441–465.Google Scholar
Glöckle, W. G. and Nonnenmacher, T. F. (1994). Fractional relaxation and the time-temperature superposition principle. Rheologica Acta 33, 337–343.Google Scholar
Gradshteyn, I. S. and Ryzhik, I. M. (2000). Table of Integrals, Series, and Products, 6th edn. Academic Press, San Diego, CA.Google Scholar
Itô, K. and McKean, H. P. Jr. (1996). Diffusion Processes and Their Sample Paths. Springer, New York.Google Scholar
Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations (North-Holland Math. Stud. 204). Elsevier, Amsterdam.Google Scholar
Laskin, N. (2003). Fractional Poisson process. Commun. Nonlinear Sci. Numer. Simul. 8, 201–213.Google Scholar
Lin, G. D. (1998). On the Mittag-Leffler distributions. J. Statist. Planning Infer. 74, 1–9.Google Scholar
Mainardi, F. (1996). Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7, 1461–1477.Google Scholar
Mainardi, F. and Pagnini, G. (2007). The role of the Fox-Wright functions in fractional sub-diffusion of distributed order. J. Comput. Appl. Math. 207, 245–257.Google Scholar
Mainardi, F., Gorenflo, R. and Scalas, E. (2004). A fractional generalization of the Poisson processes. Vietnam J. Math. 32, 53–64.Google Scholar
Mainardi, F., Mura, A., Gorenflo, R. and Stojanović, M. (2007). The two forms of fractional relaxation of distributed order. J. Vibration Control 13, 1249–1268.Google Scholar
Meerschaert, M. M., Nane, E. and Veillaisamy, P. (2011). The fractional Poisson process and the inverse stable subordinator. Electron J. Prob. 16, 1600–1620.Google Scholar
Metzler, R. and Nonnenmacher, T. F. (2003). Fractional relaxation processes and fractional rheological models for the description of a class of viscoelastic materials. Internat. J. Plasticity 19, 941–959.Google Scholar
Nonnenmacher, T. F. (1991). Fractional relaxation equation for viscoelasticity and related phenomena. In Rheological Modelling (Lecture Notes Phys. 381), Springer, New York, pp. 309–320.Google Scholar
Orsingher, E. and Beghin, L. (2004). Time-fractional equations and telegraph processes with Brownian time. Prob. Theory Relat. Fields 128, 141–160.CrossRefGoogle Scholar
Orsingher, E. and Beghin, L. (2009). Fractional diffusion equations and processes with randomly varying time. Ann. Prob. 37, 206–249.Google Scholar
Orsingher, E., Polito, F. and Sakhno, L. (2010). Fractional non-linear, linear and sublinear death processes. J. Statist. Phys. 141, 68–93.Google Scholar
Pillai, R. N. (1990). On Mittag-Leffler functions and related distributions. Ann. Inst. Statist. Math. 42, 157–161.Google Scholar
Podlubny, I. (1999). Fractional Differential Equations. Academic Press, San Diego, CA.Google Scholar
Saji Kumar, V. R. and Pillai, R. N. (2006). Single server queue with batch arrivals and α-Poisson distribution. Calcutta Statist. Assoc. Bull. 58, 93–103.Google Scholar
Schiessel, H. and Blumen, A. (1993). Hierarchical analogues to fractional relaxation equations. J. Phys. A 26, 5057–5069.CrossRefGoogle Scholar
Sibatov, R. T. and Uchaikin, D. V. (2010). Fractional relaxation and wave equations for dielectrics characterized by the Havriliak-Negami response function. Preprint. Available at http://arXiv.org/abs/1008.3972v1.Google Scholar
Uchaikin, V. V. (2002). A simple stochastic model for fractional relaxation processes. J. Math. Sci. 111, 3613–3622.CrossRefGoogle Scholar
Wang, X.-T., Zhang, S.-Y. and Fan, S. (2007). Nonhomogeneous fractional Poisson processes. Chaos Solitons Fractals 31, 236–241.Google Scholar