Geodesic
Probabilistic representation for Cauchy problem solution for evolution equation with Riemann–Liouville operator
Teoriâ veroâtnostej i ee primeneniâ, Tome 61 (2016) no. 3, pp. 417-438 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper studies properties of probabilistic approximation of a solution of the Cauchy problem for evolution equations with fractional differential operators of order more than two. To this end we construct analogous one-sided $\alpha$-stable distributions for noninteger $\alpha>2$. Although densities of these distributions are signed functions, using generalized functions methods, it is possible to give them an exact probability sense.
Keywords: Liouville–Riemann operator
Mots-clés : evolution equation, stable distribution. 
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     author = {M. V. Platonova},
     title = {Probabilistic representation for {Cauchy} problem solution for evolution equation with {Riemann{\textendash}Liouville} operator},
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M. V. Platonova. Probabilistic representation for Cauchy problem solution for evolution equation with Riemann–Liouville operator. Teoriâ veroâtnostej i ee primeneniâ, Tome 61 (2016) no. 3, pp. 417-438. http://geodesic.mathdoc.fr/item/TVP_2016_61_3_a0/

[1] Ibragimov I. A., Smorodina N. V., Faddeev M. M., “Kompleksnyi analog tsentralnoi predelnoi teoremy i veroyatnostnaya approksimatsiya integrala Feinmana”, Dokl. RAN, 459:4 (2014), 400–402 | DOI | Zbl

[2] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972, 740 pp.

[3] Kingman Dzh., Puassonovskie protsessy, MTsNMO, M., 2007, 136 pp.

[4] Kusis P., Vvedenie v teoriyu prostranstv $H_{p}$, Mir, M., 1984, 368 pp.

[5] Platonova M. V., “Simmetrichnye $\alpha$-ustoichivye raspredeleniya s netselym $\footnotesize\alpha>2$ i svyazannye s nimi stokhasticheskie protsessy”, Zap. nauchn. sem. POMI, 442, 2015, 101–117 | MR

[6] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987, 688 pp.

[7] Lachal A., “From pseudo-random walk to pseudo-Brownian motion: first exit time from a one-sided or a two-sided interval”, Int. J. Stoch. Anal., 2014 (2014), 520136, 49 | MR | Zbl

[8] Nigmatullin R. R., “The realization of the generalized transfer equation in a medium with fractal geometry”, Phys. Status. Sol., 133:1 (1986), 425–430 | DOI | MR

[9] Orsingher E., Toaldo B., “Pseudoprocesses related to space-fractional higher-order heat-type equations”, Stoch. Anal. Appl., 32:4 (2014), 619–641 | DOI | MR | Zbl

[10] Saichev A. I., Zaslavsky G. M., “Fractional kinetic equations: solutions and applications”, AIP, Chaos, 7:4 (1997), 753–764 | DOI | MR | Zbl

[11] Schwartz L., Theorie des distributions, Hermann, Paris, 1998, 418 pp. | MR | Zbl

[12] Smorodina N. V., Faddeev M. M., “The Lévy–Khinchin representation of the one class of signed stable measures and some its applications”, Acta Appl. Math., 110:3 (2010), 1289–1308 | DOI | MR | Zbl

[13] Zaslavsky G. M., “Chaos, fractional kinetics, and anomalous transport”, Phys. Repts., 371:6 (2002), 461–580 | DOI | MR | Zbl