Reflected spectrally negative stable processes and their governing equations

Baeumer, Boris; Kovács, Mihály; Meerschaert, Mark; Schilling, René; Straka, Peter

doi:10.1090/tran/6360

Reflected spectrally negative stable processes and their governing equations
HTML articles powered by AMS MathViewer

by Boris Baeumer, Mihály Kovács, Mark M. Meerschaert, René L. Schilling and Peter Straka PDF

Trans. Amer. Math. Soc. 368 (2016), 227-248 Request permission

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.

References

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, DOI 10.1023/A:1016539022492
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, DOI 10.1214/aop/1015345772
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, DOI 10.1007/978-3-0348-0087-7
Boris Baeumer and Mark M. Meerschaert, Stochastic solutions for fractional Cauchy problems, Fract. Calc. Appl. Anal. 4 (2001), no. 4, 481–500. MR 1874479
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, DOI 10.1007/s00028-009-0012-z
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, DOI 10.1239/jap/1261670691
Emilia Grigorova Bajlekova, Fractional evolution equations in Banach spaces, Eindhoven University of Technology, Eindhoven, 2001. Dissertation, Technische Universiteit Eindhoven, Eindhoven, 2001. MR 1868564
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, DOI 10.1007/s00440-009-0257-z
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, DOI 10.1214/aop/1079021462
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.
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, DOI 10.1214/10-AOP598
Jean Bertoin, Lévy processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996. MR 1406564
N. H. Bingham, Maxima of sums of random variables and suprema of stable processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 26 (1973), 273–296. MR 415780, DOI 10.1007/BF00534892
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, DOI 10.1239/jap/1165505201
F. Carlson, Une inégalité, Ark. Mat. 25B (1935), 1–5.
D. del-Castillo-Negrete, Fractional diffusion models of nonlocal transport, Phys. Plasmas 13 (2006), no. 8, 082308, 16. MR 2249732, DOI 10.1063/1.2336114
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, DOI 10.1137/110833294
R. E. Edwards, On functions which are Fourier transforms, Proc. Amer. Math. Soc. 5 (1954), 71–78. MR 60158, DOI 10.1090/S0002-9939-1954-0060158-5
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
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, DOI 10.1002/9780470316658
R. Gorenflo and F. Mainardi, Fractional diffusion processes: Probability distribution and continuous time random walk, Lecture Notes in Physics 621 (2003), 148–166.
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, DOI 10.1155/2011/298628
T. H. Hildebrandt, Introduction to the theory of integration, Pure and Applied Mathematics, Vol. XIII, Academic Press, New York-London, 1963. MR 0154957
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
K. Itô and H. P. McKean Jr., Brownian motions on a half line, Illinois J. Math. 7 (1963), 181–231. MR 154338
Niels Jacob, Pseudo-differential operators and Markov processes, Mathematical Research, vol. 94, Akademie Verlag, Berlin, 1996. MR 1409607
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
V. N. Kolokol′tsov, 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, DOI 10.1137/S0040585X97983857
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, DOI 10.1016/j.camwa.2009.08.015
Eugene Lukacs, Stable distributions and their characteristic functions, Jber. Deutsch. Math.-Verein. 71 (1969), no. Heft 2, Abt. 1, 84–114. MR 258096
M. Magdziarz and A. Weron, Competition between subdiffusion and Lévy flights: Stochastic and numerical approach, Phys. Rev. E 75 (2007), 056702.
Francesco Mainardi, Fractional calculus and waves in linear viscoelasticity, Imperial College Press, London, 2010. An introduction to mathematical models. MR 2676137, DOI 10.1142/9781848163300
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, DOI 10.1239/jap/1091543414
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, DOI 10.1016/j.cam.2004.01.033
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, DOI 10.1016/j.apnum.2005.02.008
Mark M. Meerschaert, Erkan Nane, and P. Vellaisamy, Fractional Cauchy problems on bounded domains, Ann. Probab. 37 (2009), no. 3, 979–1007. MR 2537547, DOI 10.1214/08-AOP426
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
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, DOI 10.1214/13-AOP905
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, DOI 10.1016/S0370-1573(00)00070-3
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, DOI 10.1088/0305-4470/37/31/R01
Ralf Metzler and Joseph Klafter, Boundary value problems for fractional diffusion equations, Phys. A 278 (2000), no. 1-2, 107–125. MR 1763650, DOI 10.1016/S0378-4371(99)00503-8
Pierre Patie and Thomas Simon, Intertwining certain fractional derivatives, Potential Anal. 36 (2012), no. 4, 569–587. MR 2904634, DOI 10.1007/s11118-011-9241-1
Ross G. Pinsky, Positive harmonic functions and diffusion, Cambridge Studies in Advanced Mathematics, vol. 45, Cambridge University Press, Cambridge, 1995. MR 1326606, DOI 10.1017/CBO9780511526244
Jan Prüss, Evolutionary integral equations and applications, Monographs in Mathematics, vol. 87, Birkhäuser Verlag, Basel, 1993. MR 1238939, DOI 10.1007/978-3-0348-8570-6
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. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
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
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, DOI 10.1007/3-540-28727-2_{1}
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, DOI 10.1007/s004400050201
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, DOI 10.1016/j.exmath.2009.12.002
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.
David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
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
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
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, DOI 10.1007/s10955-006-9042-x
Y. Zhang, M. M. Meerschaert, and B. Baeumer, Particle tracking for time-fractional diffusion, Phys. Rev. E 78 (2008), 036705.
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