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Reflected spectrally negative stable processes and their governing equations


Authors: Boris Baeumer, Mihály Kovács, Mark M. Meerschaert, René L. Schilling and Peter Straka
Journal: Trans. Amer. Math. Soc. 368 (2016), 227-248
MSC (2010): Primary 60G52, 60J50; Secondary 26A33, 60J22
DOI: https://doi.org/10.1090/tran/6360
Published electronically: April 20, 2015
MathSciNet review: 3413862
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Abstract: This paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one, reflected at its infimum. First we derive the forward equation using the theory of sun-dual semigroups. The resulting forward equation is a boundary value problem on the positive half-line that involves a negative Riemann-Liouville fractional derivative in space, and a fractional reflecting boundary condition at the origin. Then we apply numerical methods to explicitly compute the transition density of this space-inhomogeneous Markov process, for any starting point, to any desired degree of accuracy. Finally, we discuss an application to fractional Cauchy problems, which involve a positive Caputo fractional derivative in time.


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  • [1] Om P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynam. 29 (2002), no. 1-4, 145-155. Fractional order calculus and its applications. MR 1926471 (2003h:35271), https://doi.org/10.1023/A:1016539022492
  • [2] Hassan Allouba and Weian Zheng, Brownian-time processes: the PDE connection and the half-derivative generator, Ann. Probab. 29 (2001), no. 4, 1780-1795. MR 1880242 (2002j:60118), https://doi.org/10.1214/aop/1015345772
  • [3] Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, and Frank Neubrander, Vector-valued Laplace transforms and Cauchy problems, 2nd ed., Monographs in Mathematics, vol. 96, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2798103 (2012b:47109)
  • [4] Boris Baeumer and Mark M. Meerschaert, Stochastic solutions for fractional Cauchy problems, Fract. Calc. Appl. Anal. 4 (2001), no. 4, 481-500. MR 1874479 (2003d:26006)
  • [5] Boris Baeumer, Markus Haase, and Mihály Kovács, Unbounded functional calculus for bounded groups with applications, J. Evol. Equ. 9 (2009), no. 1, 171-195. MR 2501357 (2010g:47035), https://doi.org/10.1007/s00028-009-0012-z
  • [6] Boris Baeumer, Mark M. Meerschaert, and Erkan Nane, Space-time duality for fractional diffusion, J. Appl. Probab. 46 (2009), no. 4, 1100-1115. MR 2582709 (2011a:60178), https://doi.org/10.1239/jap/1261670691
  • [7] Emilia Grigorova Bajlekova, Fractional evolution equations in Banach spaces, Eindhoven University of Technology, Eindhoven, 2001. Dissertation, Technische Universiteit Eindhoven, Eindhoven, 2001. MR 1868564 (2002h:34115)
  • [8] Martin T. Barlow and Jiří Černý, Convergence to fractional kinetics for random walks associated with unbounded conductances, Probab. Theory Related Fields 149 (2011), no. 3-4, 639-673. MR 2776627 (2012d:60091), https://doi.org/10.1007/s00440-009-0257-z
  • [9] Peter Becker-Kern, Mark M. Meerschaert, and Hans-Peter Scheffler, Limit theorems for coupled continuous time random walks, Ann. Probab. 32 (2004), no. 1B, 730-756. MR 2039941 (2004m:60092), https://doi.org/10.1214/aop/1079021462
  • [10] D. A. Benson and M. M. Meerschaert, A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations, Adv. Water Resour. 32 (2009), 532-539.
  • [11] Violetta Bernyk, Robert C. Dalang, and Goran Peskir, Predicting the ultimate supremum of a stable Lévy process with no negative jumps, Ann. Probab. 39 (2011), no. 6, 2385-2423. MR 2932671, https://doi.org/10.1214/10-AOP598
  • [12] Jean Bertoin, Lévy processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996. MR 1406564 (98e:60117)
  • [13] N. H. Bingham, Maxima of sums of random variables and suprema of stable processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 26 (1973), 273-296. MR 0415780 (54 #3859)
  • [14] M. E. Caballero and L. Chaumont, Conditioned stable Lévy processes and the Lamperti representation, J. Appl. Probab. 43 (2006), no. 4, 967-983. MR 2274630 (2008d:60058), https://doi.org/10.1239/jap/1165505201
  • [15] F. Carlson, Une inégalité, Ark. Mat. 25B (1935), 1-5.
  • [16] D. del-Castillo-Negrete, Fractional diffusion models of nonlocal transport, Phys. Plasmas 13 (2006), no. 8, 082308, 16. MR 2249732 (2007c:76081), https://doi.org/10.1063/1.2336114
  • [17] Qiang Du, Max Gunzburger, R. B. Lehoucq, and Kun Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev. 54 (2012), no. 4, 667-696. MR 3023366, https://doi.org/10.1137/110833294
  • [18] R. E. Edwards, On functions which are Fourier transforms, Proc. Amer. Math. Soc. 5 (1954), 71-78. MR 0060158 (15,633a)
  • [19] Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989 (2000i:47075)
  • [20] Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085 (88a:60130)
  • [21] R. Gorenflo and F. Mainardi, Fractional diffusion processes: Probability distribution and continuous time random walk, Lecture Notes in Physics 621 (2003), 148-166.
  • [22] H. J. Haubold, A. M. Mathai, and R. K. Saxena, Mittag-Leffler functions and their applications, J. Appl. Math. , posted on (2011), Art. ID 298628, 51. MR 2800586 (2012e:33061), https://doi.org/10.1155/2011/298628
  • [23] T. H. Hildebrandt, Introduction to the theory of integration, Pure and Applied Mathematics, Vol. XIII, Academic Press, New York-London, 1963. MR 0154957 (27 #4900)
  • [24] Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR 0089373 (19,664d)
  • [25] K. Itô and H. P. McKean Jr., Brownian motions on a half line, Illinois J. Math. 7 (1963), 181-231. MR 0154338 (27 #4287)
  • [26] Niels Jacob, Pseudo-differential operators and Markov processes, Mathematical Research, vol. 94, Akademie Verlag, Berlin, 1996. MR 1409607 (97m:60109)
  • [27] Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. MR 2218073 (2007a:34002)
  • [28] V. N. Kolokoltsov, Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics, Teor. Veroyatn. Primen. 53 (2008), no. 4, 684-703 (Russian, with Russian summary); English transl., Theory Probab. Appl. 53 (2009), no. 4, 594-609. MR 2766141 (2011j:60151), https://doi.org/10.1137/S0040585X97983857
  • [29] Yury Luchko, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Comput. Math. Appl. 59 (2010), no. 5, 1766-1772. MR 2595950 (2010k:35515), https://doi.org/10.1016/j.camwa.2009.08.015
  • [30] Eugene Lukacs, Stable distributions and their characteristic functions, Jber. Deutsch. Math.-Verein. 71 (1969), no. Heft 2, Abt. 1, 84-114. MR 0258096 (41 #2743)
  • [31] M. Magdziarz and A. Weron, Competition between subdiffusion and Lévy flights: Stochastic and numerical approach, Phys. Rev. E 75 (2007), 056702.
  • [32] Francesco Mainardi, Fractional calculus and waves in linear viscoelasticity, Imperial College Press, London, 2010. An introduction to mathematical models. MR 2676137 (2011e:74002)
  • [33] Mark M. Meerschaert and Hans-Peter Scheffler, Limit theorems for continuous-time random walks with infinite mean waiting times, J. Appl. Probab. 41 (2004), no. 3, 623-638. MR 2074812 (2005f:60105)
  • [34] Mark M. Meerschaert and Charles Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math. 172 (2004), no. 1, 65-77. MR 2091131 (2005h:65138), https://doi.org/10.1016/j.cam.2004.01.033
  • [35] Mark M. Meerschaert and Charles Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math. 56 (2006), no. 1, 80-90. MR 2186432 (2006j:65244), https://doi.org/10.1016/j.apnum.2005.02.008
  • [36] Mark M. Meerschaert, Erkan Nane, and P. Vellaisamy, Fractional Cauchy problems on bounded domains, Ann. Probab. 37 (2009), no. 3, 979-1007. MR 2537547 (2010h:60121), https://doi.org/10.1214/08-AOP426
  • [37] Mark M. Meerschaert and Alla Sikorskii, Stochastic models for fractional calculus, de Gruyter Studies in Mathematics, vol. 43, Walter de Gruyter & Co., Berlin, 2012. MR 2884383
  • [38] Mark M. Meerschaert and Peter Straka, Semi-Markov approach to continuous time random walk limit processes, Ann. Probab. 42 (2014), no. 4, 1699-1723. MR 3262490, https://doi.org/10.1214/13-AOP905
  • [39] Ralf Metzler and Joseph Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), no. 1, 77. MR 1809268 (2001k:82082), https://doi.org/10.1016/S0370-1573(00)00070-3
  • [40] Ralf Metzler and Joseph Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37 (2004), no. 31, R161-R208. MR 2090004, https://doi.org/10.1088/0305-4470/37/31/R01
  • [41] Ralf Metzler and Joseph Klafter, Boundary value problems for fractional diffusion equations, Phys. A 278 (2000), no. 1-2, 107-125. MR 1763650 (2001b:35138), https://doi.org/10.1016/S0378-4371(99)00503-8
  • [42] Pierre Patie and Thomas Simon, Intertwining certain fractional derivatives, Potential Anal. 36 (2012), no. 4, 569-587. MR 2904634, https://doi.org/10.1007/s11118-011-9241-1
  • [43] Ross G. Pinsky, Positive harmonic functions and diffusion, Cambridge Studies in Advanced Mathematics, vol. 45, Cambridge University Press, Cambridge, 1995. MR 1326606 (96m:60179)
  • [44] Jan Prüss, Evolutionary integral equations and applications, Monographs in Mathematics, vol. 87, Birkhäuser Verlag, Basel, 1993. MR 1238939 (94h:45010)
  • [45] Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikolskiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689 (96d:26012)
  • [46] Ken-iti Sato, Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original; Revised by the author. MR 1739520 (2003b:60064)
  • [47] Enrico Scalas, Five years of continuous-time random walks in econophysics, The complex networks of economic interactions, Lecture Notes in Econom. and Math. Systems, vol. 567, Springer, Berlin, 2006, pp. 3-16. MR 2198447, https://doi.org/10.1007/3-540-28727-2_1
  • [48] René L. Schilling, Growth and Hölder conditions for the sample paths of Feller processes, Probab. Theory Related Fields 112 (1998), no. 4, 565-611. MR 1664705 (99m:60131), https://doi.org/10.1007/s004400050201
  • [49] Thomas Simon, Fonctions de Mittag-Leffler et processus de Lévy stables sans sauts négatifs, Expo. Math. 28 (2010), no. 3, 290-298 (French, with English and French summaries). MR 2671005 (2011g:60090), https://doi.org/10.1016/j.exmath.2009.12.002
  • [50] V.R. Voller (2010) An exact solution of a limit case Stefan problem governed by a fractional diffusion equation. Int. J. Heat Mass Trans. 53, 5622-5625.
  • [51] David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923 (3,232d)
  • [52] Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913 (82i:46002)
  • [53] A. H. Zemanian, Distribution theory and transform analysis, 2nd ed., Dover Publications, Inc., New York, 1987. An introduction to generalized functions, with applications. MR 918977 (88h:46081)
  • [54] Zhang Yong, David A. Benson, Mark M. Meerschaert, and Hans-Peter Scheffler, On using random walks to solve the space-fractional advection-dispersion equations, J. Stat. Phys. 123 (2006), no. 1, 89-110. MR 2225237 (2006k:82142), https://doi.org/10.1007/s10955-006-9042-x
  • [55] Y. Zhang, M. M. Meerschaert, and B. Baeumer, Particle tracking for time-fractional diffusion, Phys. Rev. E 78 (2008), 036705.
  • [56] V. M. Zolotarev, Expression of the density of a stable distribution with exponent $ \alpha $ greater than one by means of a density with exponent $ 1/\alpha $, Dokl. Akad. Nauk SSSR (N.S.) 98 (1954), 735-738 (Russian). MR 0065840 (16,493i)

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Additional Information

Boris Baeumer
Affiliation: Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand
Email: bbaeumer@maths.otago.ac.nz

Mihály Kovács
Affiliation: Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand
Email: mkovacs@maths.otago.ac.nz

Mark M. Meerschaert
Affiliation: Department of Probability and Statistics, Michigan State University, East Lansing, Michigan 48824
Email: mcubed@stt.msu.edu

René L. Schilling
Affiliation: Institut für Mathematische Stochastik, Technische Universität Dresden, 01069 Dresden, Germany
Email: rene.schilling@tu-dresden.de

Peter Straka
Affiliation: School of Mathematics and Statistics, The University of New South Wales, Kensington NSW 2052, Australia
Email: p.straka@unsw.edu.au

DOI: https://doi.org/10.1090/tran/6360
Keywords: Stable process, reflecting boundary condition, Markov process, fractional derivative, Cauchy problem
Received by editor(s): February 15, 2013
Received by editor(s) in revised form: August 6, 2013, and October 29, 2013
Published electronically: April 20, 2015
Additional Notes: The third author was partially supported by NSF grants DMS-1025486 and DMS-0803360, and NIH grant R01-EB012079.
Article copyright: © Copyright 2015 American Mathematical Society

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