Censored stable subordinators and fractional derivatives

References

[1] M.T. Barlow, J. Černý, Convergence to fractional kinetics for random walks associated with unbounded conductances. Probability Theory and Related Fields 149, No 3 (2011), 639–673; DOI: 10.1007/s00440-009-0257-z.Search in Google Scholar

[2] P. Becker-Kern, M.M. Meerschaert, H.-P. Scheffler, Limit theorems for coupled continuous time random walks. Ann. Probab. 32, No 1B (2004), 730–756; DOI: 10.1214/aop/1079021462.Search in Google Scholar

[3] V. Bernyk, R.C. Dalang, G. Peskir, Predicting the ultimate supremum of a stable Levy process with no negative jumps. Ann. Probab. 39, No 6 (2011), 2385–2423; DOI: 10.1214/10-AOP598.Search in Google Scholar

[4] J. Bertoin, Lévy Processes. Cambridge University Press, Cambridge (1996).Search in Google Scholar

[5] N.H. Bingham, Limit theorems for occupation times of Markov processes. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 17, No 1 (1971), 1–22; DOI: 10.1007/BF00538470.Search in Google Scholar

[6] K. Bogdan, K. Burdzy, Z.-Q. Chen, Censored stable processes. Probability Theory and Related Fields 127, No 1 (2003), 89–152; DOI: 10.1007/s00440-003-0275-1.Search in Google Scholar

[7] K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song, Z. Vondraček, Potential Analysis of Stable Processes and its Extensions. Springer, Berlin-Heidelberg (2009).10.1007/978-3-642-02141-1Search in Google Scholar

[8] L. Bondesson, G.K. Kristiansen, F.W. Steutel, Infinite divisibility of random variables and their integer parts. Statistics & Probability Letters 28, No 3 (1996), 271–278; DOI: 10.1016/0167-7152(95)00135-2.Search in Google Scholar

[9] B. Böttcher, R. Schilling, J. Wang, Lévy Matters III, Lévy-Type Processes, Construction, Approximation and Sample Path Properties. Springer International Publishing, Cham (2013).10.1007/978-3-319-02684-8_5Search in Google Scholar

[10] L. Boudabsa, T. Simon, P. Vallois, Fractional extreme distributions. arXiv e-prints, arXiv:1908.00584 (2019).10.1214/20-EJP520Search in Google Scholar

[11] A. Carpinteri, F. Mainardi, Fractals and fractional calculus in continuum mechanics. In: International Centre for Mechanical Sciences, Springer, Vienna (1997).Search in Google Scholar

[12] Z.-Q. Chen, M.M. Meerschaert, E. Nane, Space–time fractional diffusion on bounded domains. Journal of Mathematical Analysis and Applications 393, No 2 (2012), 479–488; DOI: 10.1016/j.jmaa.2012.04.032.Search in Google Scholar

[13] K.L. Chung, Z. Zhao, From Brownian Motion to Schrödinger's Equation. Springer, Berlin-Heidelberg (1995).10.1007/978-3-642-57856-4Search in Google Scholar

[14] K. Diethelm, The Analysis of Fractional Differential Equations, An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin-Heidelberg (2010).Search in Google Scholar

[15] Q. Du, Nonlocal Modeling, Analysis, and Computation. Ser. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (2019).Search in Google Scholar

[16] Q. Du, L. Toniazzi, Z. Xu, Censored stable subordinators and fractional derivatives. arXiv e-prints, arXiv:1906.07296 (2021).10.1515/fca-2021-0045Search in Google Scholar

[17] Q. Du, L. Toniazzi, Z. Zhou, Stochastic representation of solution to nonlocal-in-time diffusion. Stochastic Processes and their Applications 130, No 4 (2020), 2058–2085; DOI: 10.1016/j.spa.2019.06.011.Search in Google Scholar

[18] Q. Du, J. Yang, Z. Zhou, Analysis of a nonlocal-in-time parabolic equation. Discrete & Continuous Dynamical Systems - B 22, No 2 (2016), 339–368; DOI: 10.3934/dcdsb.2017016.Search in Google Scholar

[19] E.B. Dynkin, Markov Processes, Vol. 1. Springer, Berlin-Heidelberg (1965).10.1007/978-3-662-00031-1_1Search in Google Scholar

[20] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2. John Wiley & Sons, New York (1971).Search in Google Scholar

[21] R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin-Heidelberg, 2nd Ed. (2020).10.1007/978-3-662-61550-8Search in Google Scholar

[22] G. Grubb, Fractional-order operators: Boundary problems, heat equations. In: Mathematical Analysis and Applications–Plenary Lectures, Springer, Cham (2017), 51–81.10.1007/978-3-030-00874-1_2Search in Google Scholar

[23] M. Hairer, G. Iyer, L. Koralov, A. Novikov, Z. Pajor-Gyulai, A fractional kinetic process describing the intermediate time behaviour of cellular flows. Ann. Probab. 46, No 2 (2018), 897–955; DOI: 10.1214/17-AOP1196.Search in Google Scholar

[24] M.E. Hernández-Hernández, V.N. Kolokoltsov, On the probabilistic approach to the solution of generalized fractional differential equations of Caputo and Riemann-Liouville type. Journal of Fractional Calculus and Applications 7, No 1 (2016), 147–175.Search in Google Scholar

[25] M.E. Hernández-Hernández, V.N. Kolokoltsov, L. Toniazzi, Generalised fractional evolution equations of Caputo type. Chaos, Solitons & Fractals 102 (2017), 184–196; DOI: 10.1016/j.chaos.2017.05.005.Search in Google Scholar

[26] N. Ikeda, M. Nagasawa, S. Watanabe, A construction of markov processes by piecing out. Proc. Japan Acad. 42, No 4 (1966), 370–375; DOI: 10.3792/pja/1195522037.Search in Google Scholar

[27] N. Ikeda, M. Nagasawa, S. Watanabe, Fundamental equations of branching Markov processes. Proc. Japan Acad. 42, No 3 (1966), 252–257; DOI: 10.3792/pja/1195522086.Search in Google Scholar

[28] N. Jacob, Pseudo Differential Operators & Markov Processes, Vol. 3. Imperial College Press (2005).10.1142/p395Search in Google Scholar

[29] N. Jacob, A.M. Krägeloh, The Caputo derivative, Feller semigroups, and the fractional power of the first order derivative on C∞(R0+) . Fract. Calc. Appl. Anal. 5, No 4 (2002), 395–410.Search in Google Scholar

[30] A.K. Jonscher, A. Jurlewicz, K. Weron, Stochastic schemes of dielectric relaxation in correlated-cluster systems. arXiv e-prints, arXiv:cond-mat/0210481 (2002).Search in Google Scholar

[31] E. Jum, K. Kobayashi, A strong and weak approximation scheme for stochastic differential equations driven by a time-changed Brownian motion. Probability and Mathematical Statistics 36, No 2 (2016), 201–220.Search in Google Scholar

[32] Y.P. Kalmykov, W.T. Coffey, D.S.F. Crothers, S.V. Titov, Microscopic models for dielectric relaxation in disordered systems. Physical Review E 70, No 4 (2004), # 041103; DOI: 10.1103/PhysRevE.70.041103.Search in Google Scholar PubMed

[33] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Ser. North-Holland Mathematics Studies, Vol. 204, North-Holland (2006).Search in Google Scholar

[34] V.N. Kolokoltsov, The probabilistic point of view on the generalized fractional partial differential equations. Fract. Calc. Appl. Anal. 22, No 3 (2019), 543–600; DOI: 10.1515/fca-2019-0033; https://www.degruyter.com/journal/key/FCA/22/3/html.Search in Google Scholar

[35] M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal. 20, No 1 (2017), 7–51; DOI: 10.1515/fca-2017-0002; https://www.degruyter.com/journal/key/FCA/20/1/html.Search in Google Scholar

[36] A.E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin-Heidelberg (2006).Search in Google Scholar

[37] A.E. Kyprianou, J.C. Pardo, A.R. Watson, Hitting distributions of α-stable processes via path censoring and self-similarity. Ann. Probab. 42, No 1 (2014), 398–430; DOI: 10.1214/12-AOP790.Search in Google Scholar

[38] K. Lätt, A. Pedas, G. Vainikko, A smooth solution of a singular fractional differential equation. Zeitschrift fuer Analysis und Ihre Anwendungen 34 (2015), 127–147.10.4171/ZAA/1532Search in Google Scholar

[39] A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M.M. Meerschaert, M. Ainsworth, G.E. Karniadakis, What is the fractional Laplacian? A comparative review with new results. Journal of Computational Physics 404 (2020), # 109009; DOI: 10.1016/j.jcp.2019.109009.Search in Google Scholar

[40] M.M. Meerschaert, H.-P. Scheffler, Limit theorems for continuous-time random walks with infinite mean waiting times. Journal of Applied Probability 41, No 3 (2004), 623–638.10.1239/jap/1091543414Search in Google Scholar

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

[42] R. Metzler, J. Klafter, The restaurant at the end of the random walk, recent developments in the description of anomalous transport by fractional dynamics. Journal of Physics A, Mathematical and General 37, No 31 (2004), R161–R208; DOI: 101088/0305-4470/37/31/R01.Search in Google Scholar

[43] E.C. de Oliveira, F. Mainardi, J. Vaz, Fractional models of anomalous relaxation based on the Kilbas and Saigo function. Meccanica 49, No 9 (2014), 2049–2060; DOI: 10.1007/s11012-014-9930-0.Search in Google Scholar

[44] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Ser. Mathematics in Science and Engineering, Vol. 198, Elsevier (1999).Search in Google Scholar

[45] R.L. Schilling, R. Song, Z. Vondraček, Bernstein Functions. De Gruyter, Berlin (2012).10.1515/9783110269338Search in Google Scholar

[46] T. Simon, Comparing Fréchet and positive stable laws. Electron. J. Probab. 19 (2014); DOI: 10.1214/EJP.v19-3058.Search in Google Scholar

[47] S. Umarov, Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols. Springer International Publishing (2015); DOI: 10.1007/978-3-319-20771-1.Search in Google Scholar

[48] G. Vainikko, Which functions are fractionally differentiable? Zeitschrift fuer Analysis und Ihre Anwendungen 35 (2016), 465–487; DOI: 10.4171/ZAA/1574.Search in Google Scholar

[49] K. Weron, A. Jurlewicz, M. Magdziarz, A. Weron, J. Trzmiel, Overshooting and undershooting subordination scenario for fractional two-power-law relaxation responses. Physical Review E 81, No 4 (2010), # 041123; DOI: 10.1103/PhysRevE.81.041123.Search in Google Scholar PubMed

[50] K. Weron, A. Stanislavsky, A. Jurlewicz, M.M. Meerschaert, H.-P. Scheffler, Clustered continuous-time random walks, diffusion and relaxation consequences. Proc. of the Royal Society A, Mathematical, Physical and Engineering Sciences 468, No 2142 (2012), 1615–1628; DOI: 10.1098/rspa.2011.0697.Search in Google Scholar PubMed PubMed Central

[51] V.M. Zolotarev, One-dimensional stable distributions. In: Translations of Mathematical Monographs, Vol. 65, American Mathematical Society (1986).10.1090/mmono/065Search in Google Scholar