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Asymptotic properties of Brownian motion delayed by inverse subordinators


Authors: Marcin Magdziarz and René L. Schilling
Journal: Proc. Amer. Math. Soc. 143 (2015), 4485-4501
MSC (2010): Primary 60G17; Secondary 60G52
DOI: https://doi.org/10.1090/proc/12588
Published electronically: April 6, 2015
MathSciNet review: 3373947
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Abstract: We study the asymptotic behaviour of the time-changed stochastic process $ \vphantom {X}^f\!X(t)=B(\vphantom {S}^f\!S (t))$, where $ B$ is a standard one-dimensional Brownian motion and $ \vphantom {S}^f\!S$ is the (generalized) inverse of a subordinator, i.e. the first-passage time process corresponding to an increasing Lévy process with Laplace exponent $ f$. This type of processes plays an important role in statistical physics in the modeling of anomalous subdiffusive dynamics. The main result of the paper is the proof of the mixing property for the sequence of stationary increments of a subdiffusion process. We also investigate various martingale properties, derive a generalized Feynman-Kac formula, the laws of large numbers and of the iterated logarithm for $ \vphantom {X}^f\!X$.


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  • [1] Boris Baeumer, Mark M. Meerschaert, and Jeff Mortensen, Space-time fractional derivative operators, Proc. Amer. Math. Soc. 133 (2005), no. 8, 2273-2282. MR 2138870 (2006m:47076), https://doi.org/10.1090/S0002-9939-05-07949-9
  • [2] Boris Baeumer, Mark M. Meerschaert, and Erkan Nane, Brownian subordinators and fractional Cauchy problems, Trans. Amer. Math. Soc. 361 (2009), no. 7, 3915-3930. MR 2491905 (2010f:60233), https://doi.org/10.1090/S0002-9947-09-04678-9
  • [3] 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
  • [4] G. Bel and E. Barkai, Weak ergodicity breaking in the continuous time random walk, Phys. Rev. Lett. 94 (2005), 240602.
  • [5] Jean Bertoin, Lévy processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996. MR 1406564 (98e:60117)
  • [6] Jean Bertoin, Iterated Brownian motion and stable $ (\frac 14)$ subordinator, Statist. Probab. Lett. 27 (1996), no. 2, 111-114. MR 1399993 (97e:60134), https://doi.org/10.1016/0167-7152(95)00051-8
  • [7] J. Bertoin, K. van Harn, and F. W. Steutel, Renewal theory and level passage by subordinators, Statist. Probab. Lett. 45 (1999), no. 1, 65-69. MR 1718352 (2001b:60106), https://doi.org/10.1016/S0167-7152(99)00043-7
  • [8] Krzysztof Burdzy, Some path properties of iterated Brownian motion, Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992) Progr. Probab., vol. 33, Birkhäuser Boston, Boston, MA, 1993, pp. 67-87. MR 1278077 (95c:60075)
  • [9] S. Cambanis, K. Podgórski, and A. Weron, Chaotic behavior of infinitely divisible processes, Studia Math. 115 (1995), no. 2, 109-127. MR 1347436 (96g:60049)
  • [10] Stamatis Cambanis, Clyde D. Hardin Jr., and Aleksander Weron, Ergodic properties of stationary stable processes, Stochastic Process. Appl. 24 (1987), no. 1, 1-18. MR 883599 (88m:60037), https://doi.org/10.1016/0304-4149(87)90024-X
  • [11] Á. Cartea, D. del-Castillo-Negrete, Fluid limit of the continuous-time random walk with general Lévy jump distribution functions, Phys. Rev. E, 76 (2007), 041105.
  • [12] A.V. Chechkin, R. Gorenflo, and I.M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed order fractional diffusion equations, Phys. Rev. E, 66 (2002), 1-7.
  • [13] William Feller, An introduction to probability theory and its applications. Vol. II., Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403 (42 #5292)
  • [14] Bert E. Fristedt and William E. Pruitt, Lower functions for increasing random walks and subordinators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 18 (1971), 167-182. MR 0292163 (45 #1250)
  • [15] Janusz Gajda and Marcin Magdziarz, Fractional Fokker-Planck equation with tempered $ \alpha $-stable waiting times: Langevin picture and computer simulation, Phys. Rev. E (3) 82 (2010), no. 1, 011117, 6. MR 2736365 (2011g:82089), https://doi.org/10.1103/PhysRevE.82.011117
  • [16] I. Golding and E.C. Cox, Physical nature of bacterial cytoplasm, Phys. Rev. Lett., 96 (2006), 098102.
  • [17] K. van Harn and F. W. Steutel, Stationarity of delayed subordinators, Stoch. Models 17 (2001), no. 3, 369-374. MR 1850584 (2002g:60050), https://doi.org/10.1081/STM-100002278
  • [18] Y. Hu, D. Pierre-Loti-Viaud, and Z. Shi, Laws of the iterated logarithm for iterated Wiener processes, J. Theoret. Probab. 8 (1995), no. 2, 303-319. MR 1325853 (96b:60073), https://doi.org/10.1007/BF02212881
  • [19] Aleksander Janicki and Aleksander Weron, Simulation and chaotic behavior of $ \alpha $-stable stochastic processes, Monographs and Textbooks in Pure and Applied Mathematics, vol. 178, Marcel Dekker, Inc., New York, 1994. MR 1306279 (96g:60026)
  • [20] J.-H. Jeon et al, In vivo anomalous diffusion and weak ergodicity breaking of lipid granules, Phys. Rev. Lett., 106 (2011), 048103.
  • [21] A. K. Jonscher, A. Jurlewicz and K. Weron, Stochastic schemes of dielectric relaxation in correlated-cluster systems, Contemp. Physics, 44 (2003), 329-339.
  • [22] A. Jurlewicz, K. Weron, and M. Teuerle, Generalized Mittag-Leffler relaxation: Clustering-jump continuous-time random walk approach, Phys. Rev. E, 78, (2008), 011103.
  • [23] I. Kaj, A. Martin-Löf, Scaling limit results for the sum of many inverse Lévy subordinators. Preprint, Institut Mittag-Leffler, (2005).
  • [24] Anatoly N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl. 340 (2008), no. 1, 252-281. MR 2376152 (2009i:35177), https://doi.org/10.1016/j.jmaa.2007.08.024
  • [25] N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. MR 0350027 (50 #2520)
  • [26] Davar Khoshnevisan and Thomas M. Lewis, Stochastic calculus for Brownian motion on a Brownian fracture, Ann. Appl. Probab. 9 (1999), no. 3, 629-667. MR 1722276 (2001m:60128), https://doi.org/10.1214/aoap/1029962807
  • [27] Andreas Nordvall Lagerås, A renewal-process-type expression for the moments of inverse subordinators, J. Appl. Probab. 42 (2005), no. 4, 1134-1144. MR 2203828 (2007c:60089), https://doi.org/10.1239/jap/1134587822
  • [28] Andrzej Lasota and Michael C. Mackey, Chaos, fractals, and noise, 2nd ed., Applied Mathematical Sciences, vol. 97, Springer-Verlag, New York, 1994. Stochastic aspects of dynamics. MR 1244104 (94j:58102)
  • [29] Marcin Magdziarz, Stochastic representation of subdiffusion processes with time-dependent drift, Stochastic Process. Appl. 119 (2009), no. 10, 3238-3252. MR 2568272 (2011d:60128), https://doi.org/10.1016/j.spa.2009.05.006
  • [15] Marcin Magdziarz, Langevin picture of subdiffusion with infinitely divisible waiting times, J. Stat. Phys. 135 (2009), no. 4, 763-772. MR 2546742 (2011a:82073), https://doi.org/10.1007/s10955-009-9751-z
  • [31] Marcin Magdziarz, Path properties of subdiffusion--a martingale approach, Stoch. Models 26 (2010), no. 2, 256-271. MR 2739351 (2011k:60129), https://doi.org/10.1080/15326341003756379
  • [32] Marcin Magdziarz and Aleksander Weron, Ergodic properties of anomalous diffusion processes, Ann. Physics 326 (2011), no. 9, 2431-2443. MR 2825444 (2012g:60113), https://doi.org/10.1016/j.aop.2011.04.015
  • [33] M. Magdziarz, A. Weron and K. Weron, Fractional Fokker-Planck dynamics: Stochastic representation and computer simulation, Phys. Rev. E, 75 (2007), 016708.
  • [34] G. Maruyama, Infinitely divisible processes, Teor. Verojatnost. i Primenen. 15 (1970), 3-23 (English, with Russian summary). MR 0285046 (44 #2270)
  • [35] Mark M. Meerschaert, David A. Benson, Hans-Peter Scheffler, and Boris Baeumer, Stochastic solution of space-time fractional diffusion equations, Phys. Rev. E (3) 65 (2002), no. 4, 041103, 4. MR 1917983 (2003d:60165), https://doi.org/10.1103/PhysRevE.65.041103
  • [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, Erkan Nane, and P. Vellaisamy, Transient anomalous sub-diffusion on bounded domains, Proc. Amer. Math. Soc. 141 (2013), no. 2, 699-710. MR 2996975, https://doi.org/10.1090/S0002-9939-2012-11362-0
  • [38] Mark M. Meerschaert, Erkan Nane, and P. Vellaisamy, Distributed-order fractional diffusions on bounded domains, J. Math. Anal. Appl. 379 (2011), no. 1, 216-228. MR 2776466 (2012e:35263), https://doi.org/10.1016/j.jmaa.2010.12.056
  • [39] Mark M. Meerschaert, Erkan Nane, and Yimin Xiao, Large deviations for local time fractional Brownian motion and applications, J. Math. Anal. Appl. 346 (2008), no. 2, 432-445. MR 2431539 (2009m:60065), https://doi.org/10.1016/j.jmaa.2008.05.087
  • [40] 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)
  • [41] Mark M. Meerschaert and Hans-Peter Scheffler, Triangular array limits for continuous time random walks, Stochastic Process. Appl. 118 (2008), no. 9, 1606-1633. MR 2442372 (2010b:60135), https://doi.org/10.1016/j.spa.2007.10.005
  • [42] M. M. Meerschaert, Y. Zhang, and B. Baeumer, Tempered anomalous diffusion in heterogeneous systems, Geophys. Res. Lett., 35 (2008), L17403.
  • [43] R. Metzler, E. Barkai and J. Klafter, Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach, Phys. Rev. Lett., 82 (1999), 3563-3567.
  • [44] 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
  • [45] Jebessa B. Mijena and Erkan Nane, Strong analytic solutions of fractional Cauchy problems, Proc. Amer. Math. Soc. 142 (2014), no. 5, 1717-1731. MR 3168478, https://doi.org/10.1090/S0002-9939-2014-11905-8
  • [46] Erkan Nane, Laws of the iterated logarithm for a class of iterated processes, Statist. Probab. Lett. 79 (2009), no. 16, 1744-1751. MR 2566748 (2010i:60100), https://doi.org/10.1016/j.spl.2009.04.013
  • [47] Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357 (2000h:60050)
  • [48] Jan Rosiński, Tempering stable processes, Stochastic Process. Appl. 117 (2007), no. 6, 677-707. MR 2327834 (2008g:60146), https://doi.org/10.1016/j.spa.2006.10.003
  • [49] Jan Rosiński and Tomasz Żak, Simple conditions for mixing of infinitely divisible processes, Stochastic Process. Appl. 61 (1996), no. 2, 277-288. MR 1386177 (97d:60126), https://doi.org/10.1016/0304-4149(95)00083-6
  • [50] A. Piryatinska, A. I. Saichev and W.A. Woyczynski, Models of anomalous diffusion: The subdiffusive case, Phys. A, 349 (2005), 375-420.
  • [51] H. Scher, G. Margolin, R. Metzler, J. Klafter, and B. Berkowitz, The dynamical foundation of fractal stream chemistry: The origin of extremely long retention times, Geophys. Res. Lett., 29 (2002), 1061.
  • [52] S. G. Samko, A. A. Kilbas and D. I. Marichev, Integrals and Derivatives of the Fractional Order and Some of Their Applications, Gordon and Breach, Amsterdam, 1993.
  • [53] 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)
  • [54] René L. Schilling, Renming Song, and Zoran Vondraček, Bernstein functions, de Gruyter Studies in Mathematics, vol. 37, Walter de Gruyter & Co., Berlin, 2010. Theory and applications. MR 2598208 (2011d:60060)
  • [55] I. M. Sokolov and J. Klafter, Field-induced dispersion in subdiffusion, Phys. Rev. Lett., 97 (2006), 140602.
  • [56] Aleksander Stanislavsky, Karina Weron, and Aleksander Weron, Diffusion and relaxation controlled by tempered $ \alpha $-stable processes, Phys. Rev. E (3) 78 (2008), no. 5, 051106, 6. MR 2551366 (2010i:82145), https://doi.org/10.1103/PhysRevE.78.051106
  • [57] Matthias Winkel, Electronic foreign-exchange markets and passage events of independent subordinators, J. Appl. Probab. 42 (2005), no. 1, 138-152. MR 2144899 (2006b:60102)
  • [58] V. M. Zolotarev, One-dimensional stable distributions, Translations of Mathematical Monographs, vol. 65, American Mathematical Society, Providence, RI, 1986. Translated from the Russian by H. H. McFaden; Translation edited by Ben Silver. MR 854867 (87k:60002)

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Marcin Magdziarz
Affiliation: Hugo Steinhaus Center, Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland
Email: marcin.magdziarz@pwr.wroc.pl

René L. Schilling
Affiliation: Technische Universität Dresden, Institut für Mathematische Stochastik, 01062 Dresden, Germany
Email: rene.schilling@tu-dresden.de

DOI: https://doi.org/10.1090/proc/12588
Received by editor(s): November 23, 2013
Received by editor(s) in revised form: July 1, 2014
Published electronically: April 6, 2015
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2015 American Mathematical Society

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