## Electronic Journal of Probability

### Pseudo-Processes Governed by Higher-Order Fractional Differential Equations

Luisa Beghin

#### Abstract

We study here a heat-type differential equation of order $n$ greater than two, in the case where the time-derivative is supposed to be fractional. The corresponding solution can be described as the transition function of a pseudoprocess $\Psi _{n}$ (coinciding with the one governed by the standard, non-fractional, equation) with a time argument $\mathcal{T}_{\alpha }$ which is itself random. The distribution of $\mathcal{T}_{\alpha }$ is presented together with some features of the solution (such as analytic expressions for its moments.

#### Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 16, 467-485.

Dates
Accepted: 31 March 2008
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819089

Digital Object Identifier
doi:10.1214/EJP.v13-496

Mathematical Reviews number (MathSciNet)
MR2386739

Zentralblatt MATH identifier
1191.60039

Subjects
Primary: 60G07: General theory of processes
Secondary: 60E07: Infinitely divisible distributions; stable distributions

Rights

#### Citation

Beghin, Luisa. Pseudo-Processes Governed by Higher-Order Fractional Differential Equations. Electron. J. Probab. 13 (2008), paper no. 16, 467--485. doi:10.1214/EJP.v13-496. https://projecteuclid.org/euclid.ejp/1464819089

#### References

• Anh, V. V.; Leonenko, N. N. Scaling laws for fractional diffusion-wave equations with singular data. Statist. Probab. Lett. 48 (2000), no. 3, 239–252.
• Anh, V. V.; Leonenko, N. N. Spectral analysis of fractional kinetic equations with random data. J. Statist. Phys. 104 (2001), no. 5-6, 1349–1387.
• Anh, V. V.; Leonenko, N. N. Spectral theory of renormalized fractional random fields. Teor. Imovir. Mat. Stat. No. 66 (2002), 3–14; translation in Theory Probab. Math. Statist. No. 66 (2003), 1–13
• Agrawal, Om P.; A general solution for the fourth-order fractional diffusion-wave equation. Fract. Calc. Appl. Anal. 3 (2000), no. 1, 1–12.
• Angulo, J. M.; Ruiz-Medina, M. D.; Anh, V. V.; Grecksch, W. Fractional diffusion and fractional heat equation. Adv. in Appl. Probab. 32 (2000), no. 4, 1077–1099.
• Beghin, Luisa; Orsingher, Enzo; The telegraph process stopped at stable-distributed times and its connection with the fractional telegraph equation. Fract. Calc. Appl. Anal. 6 (2003), no. 2, 187–204.
• Beghin, L.; Orsingher, E.; The distribution of the local time for "pseudoprocesses” and its connection with fractional diffusion equations. Stochastic Process. Appl. 115 (2005), no. 6, 1017–1040.
• Beghin, L.; Orsingher, E.; Ragozina, T. Joint distributions of the maximum and the process for higher-order diffusions. Stochastic Process. Appl. 94 (2001), no. 1, 71–93.
• Beghin, Luisa; Hochberg, Kenneth J.; Orsingher, Enzo. Conditional maximal distributions of processes related to higher-order heat-type equations. Stochastic Process. Appl. 85 (2000), no. 2, 209–223.
• Bingham, N. H.; Maxima of sums of random variables and suprema of stable processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 26 (1973), 273–296.
• Daletsky Yu. L.; Integration in function spaces, In: phProgress in Mathematics, R.V. Gamkrelidze, ed., 4, 87-132.
• Daletsky Yu. L.; Fomin S. V. Generalized measures in function spaces, Theory Prob. Appl., 10, (2) 304-316.
• ,De Gregorio A.; Pseudoprocessi governati da equazioni frazionarie di ordine superiore al secondo, Tesi di Laurea, Università di Roma La Sapienza.
• Fujita, Yasuhiro; Integrodifferential equation which interpolates the heat equation and the wave equation. Osaka J. Math. 27 (1990), no. 2, 309–321.
• Gradshteyn I. S., Rhyzik I. M.; Tables of Integrals, Series and Products, Alan Jeffrey Editor, Academic Press, London.
• Hochberg, Kenneth J.; Orsingher, Enzo. The arc-sine law and its analogs for processes governed by signed and complex measures. Stochastic Process. Appl. 52 (1994), no. 2, 273-292.
• Hochberg, Kenneth J.; Orsingher, Enzo. Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations. J. Theoret. Probab. 9 (1996), no. 2, 511–532.
• Hochberg, Kenneth J.; Some properties of the distribution corresponding to the equation fracpartial upartial t% =(-1)^p+1fracpartial ^2pupartial x^2p, Soviet Math. Dokl., 1, 260-263.
• Lachal, Aimé; Distributions of sojourn time, maximum and minimum for pseudo-processes governed by higher-order heat-type equations. Electron. J. Probab. 8 (2003), no. 20, 53 pp. (electronic).
• Lachal, Aimé; First hitting time and place for pseudo-processes driven by the equation $\frac\partial{\partial t}=\pm\frac{\partial^ N}{\partial x^ N}$ subject to a linear drift. Stochastic Process. Appl. 118 (2008), no. 1, 1–27.
• Mainardi, Francesco; Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7 (1996), no. 9, 1461–1477.
• Nigmatullin R. R.; "The realization of the generalized transfer equation in a medium with fractal geometry", Phys. Stat. Sol., (b) 133, 425-430.
• Nikitin, Y.; Orsingher, E. On sojourn distributions of processes related to some higher-order heat-type equations. J. Theoret. Probab. 13 (2000), no. 4, 997–1012.
• Nishioka, Kunio; The first hitting time and place of a half-line by a biharmonic pseudo process. Japan. J. Math. (N.S.) 23 (1997), no. 2, 235–280.
• Orsingher, E.; Processes governed by signed measures connected with third-order 'heat-type' equations, Lith. Math. Journ., 31, 321-334.
• Orsinger, E., Beghion, L.; Time-fractional equations and telegraph processes with Brownian time, Probability Theory and Related Fields., 128, 141-160.
• Podlubny, Igor; Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. xxiv+340 pp. ISBN: 0-12-558840-2
• Saichev, Alexander I.; Zaslavsky, George M. Fractional kinetic equations: solutions and applications. Chaos 7 (1997), no. 4, 753–764.
• Samko, Stefan G.; Kilbas, Anatoly A.; Marichev, Oleg I. Fractional integrals and derivatives. Theory and applications. Edited and with a foreword by S. M. NikolÊ¹skiÄ­. Translated from the 1987 Russian original. Revised by the authors. Gordon and Breach Science Publishers, Yverdon, 1993. xxxvi+976 pp. ISBN: 2-88124-864-0
• Schneider, W. R., Wyss, W.; Kilbas, Anatoly A.; Marichev, Oleg I. Stable Non-Gaussian Random Processes, Chapman and Hall, New York.
• Schneider, W. R.; Wyss, W. Fractional diffusion and wave equations. J. Math. Phys. 30 (1989), no. 1, 134–144.
• Wyss, Walter. The fractional Black-Scholes equation. Fract. Calc. Appl. Anal. 3 (2000), no. 1, 51–61.