Tempered relaxation equation and related generalized stable processes

References

[1] L. Beghin, On fractional tempered stable processes and their governing differential equations. J. of Comput. Physics293 (2015), 29–39.10.1016/j.jcp.2014.05.026Suche in Google Scholar

[2] L. Beghin, Fractional diffusion-type equations with exponential and logarithmic differential operators. Stoch. Proc. Appl. 128 (2018), 2427–2447.10.1016/j.spa.2017.09.013Suche in Google Scholar

[3] L. Beghin, C. Ricciuti, Time-inhomogeneous fractional Poisson processes defined by the multistable subordinator. Stoch. Anal. Appl. 37, No 2 (2019), 171–188.10.1080/07362994.2018.1548970Suche in Google Scholar

[4] L. Beghin, C. Ricciuti, Pseudo-differential operators and related additive geometric stable processes. Markov Processes and Related Fields25 (2019), 415–444.Suche in Google Scholar

[5] S. Bochner, Harmonic Analysis and the Theory of Probability. University of California Press, California Monogr. Math. Sci., Berkeley (1955).10.1525/9780520345294Suche in Google Scholar

[6] B.D. Carter, M.D. Springer, The distribution of products, quotients and powers of independent H-function variates. SIAM J. Appl. Math. 33, No 4 (1977), 542–558.10.1137/0133036Suche in Google Scholar

[7] I. Dimovski, V. Kiryakova, The Obrechkoff integral transform: Properties and relation to a generalized fractional calculus. Numer. Funct. Anal. and Optimiz. 21, No 1-2 (2000), 121–144; 10.1080/01630560008816944.Suche in Google Scholar

[8] W. Feller, An Introduction Probability Theory and its Applications. Vol. 2 (2nd Ed.), Wiley, New York (1971).Suche in Google Scholar

[9] J. Gajda, A. Wylomanska, Time-changed Ornstein–Uhlenbeck process. J. Phys. A: Math. Theor. 48 (2015), 1–19.10.1088/1751-8113/48/13/135004Suche in Google Scholar

[10] J. Gajda, A. Kumar, A. Wylomanska, Stable Lévy process delayed by tempered stable subordinator. Statistics and Probability Letters145 (2019), 284–292.10.1016/j.spl.2018.09.008Suche in Google Scholar

[11] R. Garra, A. Giusti, F. Mainardi, G. Pagnini, Fractional relaxation with time-varying coefficient. Fract. Calc. Appl. Anal. 17, No 2 (2014), 424–439; 10.2478/s13540-014-0178-0; https://www.degruyter.com/view/journals/fca/17/2/fca.17.issue-2.xml.Suche in Google Scholar

[12] R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer-Verlag, Berlin Heidelberg (2014).10.1007/978-3-662-43930-2Suche in Google Scholar

[13] G.J. Jameson, The incomplete gamma functions. The Math. Gazette100 (548) (2016), 298–306.10.1017/mag.2016.67Suche in Google Scholar

[14] D.B. Karp, J.L. Lopez, Representations of hypergeometric functions for arbitrary parameter values and their use. J. of Approximation Theory218 (2017), 42–70.10.1016/j.jat.2017.03.004Suche in Google Scholar

[15] D.B. Karp, J.L. Lopez, An extension of the multiple Erdélyi-Kober operator and representations of the generalized hypergeometric functions. Fract. Calc. Appl. Anal. 21, No 5 (2018), 1360–1376; 10.1515/fca-2018-0071; https://www.degruyter.com/view/journals/fca/21/5/fca.21.issue-5.xml.Suche in Google Scholar

[16] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Vol. 204 of North-Holland Math. Studies, Elsevier Science B.V., Amsterdam (2006).10.1016/S0304-0208(06)80001-0Suche in Google Scholar

[17] V. Kiryakova, Generalized Fractional Calculus and Applications. Pitman Research Notes in Mathematics, Vol. 301, Longman (1994).Suche in Google Scholar

[18] A.N. Kochubei, General fractional calculus, evolution equations and renewal processes. Integr. Equat. Operator Theory71 (2011), 583–600.10.1007/s00020-011-1918-8Suche in Google Scholar

[19] V. Kolokoltsov, The probabilistic point of view on the generalized fractional partial differential equations. Fract. Calc. Appl. Anal. 22, No 3 (2019), 543–600; 10.1515/fca-2019-0033; https://www.degruyter.com/view/journals/fca/22/3/fca.22.issue-3.xml.Suche in Google Scholar

[20] A. Kumar, P. Vellaisamy, Inverse tempered stable subordinators. Statistics and Probability Letters103 (2015), 134–141.10.1016/j.spl.2015.04.010Suche in Google Scholar

[21] F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals7, No 9 (1996), 1461–1477.10.1016/0960-0779(95)00125-5Suche in Google Scholar

[22] O.I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables. Ellis Horwood, Chichester (1983).Suche in Google Scholar

[23] A.M. Mathai, R.K. Saxena, H.J. Haubold, The H-Function: Theory and Applications. Springer, New York (2010).10.1007/978-1-4419-0916-9Suche in Google Scholar

[24] M.M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus. De Gruyter, Berlin/Boston (2012).10.1515/9783110258165Suche in Google Scholar

[25] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993).Suche in Google Scholar

[26] E. Orsingher, C. Ricciuti, B. Toaldo, Time-inhomogeneous jump processes and variable order operators. Potential Analysis45, No 3 (2016), 435–461.10.1007/s11118-016-9551-4Suche in Google Scholar

[27] R.B. Paris, Incomplete Gamma Function. In: F.W. Olver, D.M. Lozier, R. Boisvert, C.W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press (2010).Suche in Google Scholar

[28] V. Pipiras, M. Taqqu, Long-Range Dependence and Self-Similarity. Cambridge University Press (2017).10.1017/CBO9781139600347Suche in Google Scholar

[29] G. Samorodnitsky, M. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman etc., New York (1994).Suche in Google Scholar

[30] R.L. Schilling, R. Song, Z. Vondracek, Bernstein Functions: Theory and Applications. Vol. 37, De Gruyter Studies in Math. Ser., Berlin (2010).Suche in Google Scholar

[31] M.D. Springer, W.E. Thompson, The distribution of products of beta, gamma and Gaussian random variables. SIAM J. Appl. Math. 18, No 4 (1970), 721–737.10.1137/0118065Suche in Google Scholar

[32] E.W. Stacy, A generalization of the gamma distribution. The Annals of Mathematical Statistics, 33, No 3 (1962), 1187–1192.10.1214/aoms/1177704481Suche in Google Scholar

[33] A. Stanislavsky, K. Weron, A. Weron, Diffusion and relaxation controlled by tempered α-stable processes. Physical Review E2008 (2008), # 051106, 1–6.10.1103/PhysRevE.78.051106Suche in Google Scholar PubMed

[34] B. Toaldo, Convolution-type derivatives, hitting-times of subordinators and time-changed C₀-semigroups. Potential Analysis42, No 1 (2015), 115–140.10.1007/s11118-014-9426-5Suche in Google Scholar

[35] P. Vellaisamy, K.K. Kataria, The I-function distribution and its extensions. Teor. Veroyatnost. i Primenen. 63, No 2 (2018), 284–305.10.4213/tvp5184Suche in Google Scholar

[36] A. Wyłomanska, Arithmetic Brownian motion subordinated by tempered stable and inverse tempered stable processes. Physica A391 (2012), 5685–5696.10.1016/j.physa.2012.05.072Suche in Google Scholar

[37] A. Wyłomanska, The tempered stable process with infinitely divisible inverse subordinators. J. Stat. Mech. Theory Exp. (2013), 1–18.10.1088/1742-5468/2013/10/P10011Suche in Google Scholar

[38] A. Wylomanska, A. Chechkin, J. Gajda, I.M. Sokolov, Codifference as a practical tool to measure interdependence. Physica A: Statistical Mechanics and its Applications421, No 1 (2015), 412–429.10.1016/j.physa.2014.11.049Suche in Google Scholar

[39] S.J. Wolfe, On moments of probability distribution functions. In: Fractional Calculus and Its Applications, B. Ross (Ed.), Lect. Notes in Math. 457, Springer, Berlin (1975), 306–316.10.1007/BFb0067116Suche in Google Scholar