×

zbMATH — the first resource for mathematics

Fractional diffusion equations and processes with randomly varying time. (English) Zbl 1173.60027
The authors consider time-fractional diffusion equations of order \(0<\nu\leq2\) and interprete their solutions as densities of the composition of various types of stochastic processes. It is proved that the solution for \(\nu=1/2^n\) corresponds to the distribution of \(n\)-times iterated Brownian motion. The explicit solutions of the fractional diffusion equation are presented for \(\nu=1/3, 2/3, 4/3.\) In particular, \(u_{2/3}(x,t)\) is given in terms of Airy functions. The relationship between the solution \(u_{\nu}(x,t)\) and the stable densities is presented. The behavior of the solution (for \(x\) varying and \(t\) fixed), which is substantially different in the two intervals \(0<\nu\leq1\) and \(1<\nu\leq2\) is analyzed.

MSC:
60J60 Diffusion processes
26A33 Fractional derivatives and integrals
60G52 Stable stochastic processes
60J65 Brownian motion
33E12 Mittag-Leffler functions and generalizations
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Allouba, H. (2002). Brownian-time processes: The PDE connection. II. And the corresponding Feynman-Kac formula. Trans. Amer. Math. Soc. 354 4627-4637 (electronic). JSTOR: · Zbl 1006.60063
[2] Allouba, H. and Zheng, W. (2001). Brownian-time processes: The PDE connection and the half-derivative generator. Ann. Probab. 29 1780-1795. · Zbl 1018.60066
[3] Angulo, J. M., Anh, V. V., McVinish, R. and Ruiz-Medina, M. D. (2005). Fractional kinetic equations driven by Gaussian or infinitely divisible noise. Adv. in Appl. Probab. 37 366-392. · Zbl 1076.60044
[4] Angulo, J. M., Ruiz-Medina, M. D., Anh, V. V. and Grecksch, W. (2000). Fractional diffusion and fractional heat equation. Adv. in Appl. Probab. 32 1077-1099. · Zbl 0986.60077
[5] Baeumer, B., Meerschaert, M. M. and Nane, E. (2007). Brownian subordinators and fractional Cauchy problems. Available at · Zbl 1186.60079
[6] Beghin, L. and Orsingher, E. (2003). The telegraph process stopped at stable-distributed times and its connection with the fractional telegraph equation. Fract. Calc. Appl. Anal. 6 187-204. · Zbl 1083.60039
[7] Beghin, L. and Orsingher, E. (2005). The distribution of the local time for “pseudoprocesses” and its connection with fractional diffusion equations. Stochastic Process. Appl. 115 1017-1040. · Zbl 1073.60076
[8] Benachour, S., Roynette, B. and Vallois, P. (1999). Explicit solutions of some fourth order partial differential equations via iterated Brownian motion. In Seminar on Stochastic Analysis, Random Fields and Applications ( Ascona , 1996). Progr. Probab. 45 39-61. Birkhäuser, Basel. · Zbl 0945.60083
[9] Buckwar, E. and Luchko, Y. (1998). Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations. J. Math. Anal. Appl. 227 81-97. · Zbl 0932.58038
[10] Burdzy, K. and San Martín, J. (1995). Iterated law of iterated logarithm. Ann. Probab. 23 1627-1643. · Zbl 0857.60035
[11] DeBlassie, R. D. (2004). Iterated Brownian motion in an open set. Ann. Appl. Probab. 14 1529-1558. · Zbl 1051.60082
[12] Engler, H. (1997). Similarity solutions for a class of hyperbolic integrodifferential equations. Differential Integral Equations 10 815-840. · Zbl 0892.45005
[13] Feller, W. (1971). An Introduction to Probability Theory and Its Applications II , 2nd ed. Wiley, New York. · Zbl 0219.60003
[14] Fujita, Y. (1990). Integrodifferential equation which interpolates the heat equation and the wave equation. II. Osaka J. Math. 27 797-804. · Zbl 0796.45010
[15] Funaki, T. (1979). Probabilistic construction of the solution of some higher order parabolic differential equation. Proc. Japan Acad. Ser. A Math. Sci. 55 176-179. · Zbl 0433.35039
[16] Gorenflo, R. and Mainardi, F. (1997). Fractional calculus: Integral and differential equations of fractional order. In Fractals and Fractional Calculus in Continuum Mechanics ( Udine , 1996). CISM Courses and Lectures 378 223-276. Springer, Vienna. · Zbl 0916.34011
[17] Gorenflo, R., Mainardi, F. and Srivastava, H. M. (1998). Special functions in fractional relaxation-oscillation and fractional diffusion-wave phenomena. In Proceedings of the Eighth International Colloquium on Differential Equations ( Plovdiv , 1997) 195-202. VSP, Utrecht. · Zbl 0921.33009
[18] Gradshteyn, I. S. and Ryzhik, I. M. (1994). Table of Integrals, Series, and Products . Academic Press, Boston, MA. · Zbl 0918.65002
[19] Hochberg, K. J. and Orsingher, E. (1996). Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations. J. Theoret. Probab. 9 511-532. · Zbl 0878.60050
[20] Khoshnevisan, D. and Lewis, T. M. (1996). The uniform modulus of continuity of iterated Brownian motion. J. Theoret. Probab. 9 317-333. · Zbl 0880.60081
[21] Lachal, A. (2003). Distributions of sojourn time, maximum and minimum for pseudo-processes governed by higher-order heat-type equations. Electron. J. Probab. 8 1-53. · Zbl 1064.60078
[22] Lebedev, N. N. (1972). Special Functions and Their Applications . Dover, New York. · Zbl 0271.33001
[23] Lukacs, E. (1969). Stable distributions and their characteristic functions. Jber. Deutsch. Math.-Verein. 71 84-114. · Zbl 0174.50002
[24] Magnus, W. and Oberhettinger, F. (1948). Formeln und Sätze Für die Speziellen Funktionen der Mathematischen Physik , 2d ed. Springer, Berlin. · Zbl 0039.29701
[25] Mainardi, F. (1994). On the initial value problem for the fractional diffusion-wave equation. In Waves and Stability in Continuous Media ( Bologna , 1993). Ser. Adv. Math. Appl. Sci. 23 246-251. World Sci. Publ., River Edge, NJ.
[26] Mainardi, F. (1995a). The time fractional diffusion-wave equation. Izv. Vyssh. Uchebn. Zaved. Radiofiz. 38 20-36.
[27] Mainardi, F. (1995b). Fractional diffusive waves in viscoelastic solids. In Nonlinear Waves in Solids (J. L. Wegner and F. R. Norwood, eds.) 93-97. ASME, Fairfield, NJ.
[28] Mainardi, F. (1996). The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9 23-28. · Zbl 0879.35036
[29] Mainardi, F. and Tomirotti, M. (1998). On a special function arising in the time fractional diffusion-wave equation. In Transform Methods and Special Functions ( P. Rusev, I. Dimovski and V. Kiryakova, eds.). Bulgarian Academy of Sciences, IMI, Sofia. · Zbl 0921.33010
[30] McKean Jr., H. P. (1963). A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2 227-235. · Zbl 0119.34701
[31] Nigmatullin, R. R. (1986). The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Stat. Sol. 133 425-430.
[32] Nigmatullin, R. R. (2006). ‘Fractional’ kinetic equations and ‘universal’ decoupling of a memory function in mesoscale region. Phys. A 363 282-298.
[33] Nigmatullin, R. R., Arbuzov, A. A., Salehli, F., Giz, A., Bayrak, I. and Catalgil-Giz, H. (2007). The first experimental confirmation of the fractional kinetics containing the complex-power-law exponents: Dielectric measurements of polymerization reactions. Phys. B 388 418-434.
[34] Nikitin, Y. and Orsingher, E. (2000). On sojourn distributions of processes related to some higher-order heat-type equations. J. Theoret. Probab. 13 997-1012. · Zbl 0993.60042
[35] Orsingher, E. and Beghin, L. (2004). Time-fractional telegraph equations and telegraph processes with Brownian time. Probab. Theory Related Fields 128 141-160. · Zbl 1049.60062
[36] Podlubny, I. (1999). Fractional Differential Equations. Mathematics in Science and Engineering 198 . Academic Press, San Diego, CA. · Zbl 0924.34008
[37] Saichev, A. I. and Zaslavsky, G. M. (1997). Fractional kinetic equations: Solutions and applications. Chaos 7 753-764. · Zbl 0933.37029
[38] Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives . Gordon and Breach Science Publishers, Yverdon. · Zbl 0818.26003
[39] Saxena, R. K., Mathai, A. M. and Haubold, H. J. (2006). Reaction-diffusion systems and non-linear waves. Astrophysics and Space Science 305 297-303. · Zbl 1105.35308
[40] Schneider, W. R. and Wyss, W. (1989). Fractional diffusion and wave equations. J. Math. Phys. 30 134-144. · Zbl 0692.45004
[41] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics . Wiley, New York. · Zbl 1170.62365
[42] Wyss, W. (1986). The fractional diffusion equation. J. Math. Phys. 27 2782-2785. · Zbl 0632.35031
[43] Zolotarev, V. M. (1986). One-Dimensional Stable Distribution . Amer. Math. Soc., Providence, RI. · Zbl 0589.60015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.