α-time fractional brownian motion: PDE connections and local times

Erkan Nane; Dongsheng Wu; Yimin Xiao

ESAIM: Probability and Statistics (2012)

  • Volume: 16, page 1-24
  • ISSN: 1292-8100

Abstract

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For 0 < α ≤ 2 and 0 < H < 1, an α-time fractional Brownian motion is an iterated process Z =  {Z(t) = W(Y(t)), t ≥ 0}  obtained by taking a fractional Brownian motion  {W(t), t ∈ ℝ} with Hurst index 0 < H < 1 and replacing the time parameter with a strictly α-stable Lévy process {Y(t), t ≥ 0} in ℝ independent of {W(t), t ∈ R}. It is shown that such processes have natural connections to partial differential equations and, when Y is a stable subordinator, can arise as scaling limit of randomly indexed random walks. The existence, joint continuity and sharp Hölder conditions in the set variable of the local times of a d-dimensional α-time fractional Brownian motion X = {X(t), t ∈ ℝ+} defined by X(t) = (X1(t), ..., Xd(t)), where t ≥ 0 and X1, ..., Xd are independent copies of Z, are investigated. Our methods rely on the strong local nondeterminism of fractional Brownian motion.

How to cite

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Nane, Erkan, Wu, Dongsheng, and Xiao, Yimin. "α-time fractional brownian motion: PDE connections and local times." ESAIM: Probability and Statistics 16 (2012): 1-24. <http://eudml.org/doc/273606>.

@article{Nane2012,
abstract = {For 0 &lt; α ≤ 2 and 0 &lt; H &lt; 1, an α-time fractional Brownian motion is an iterated process Z =  \{Z(t) = W(Y(t)), t ≥ 0\}  obtained by taking a fractional Brownian motion  \{W(t), t ∈ ℝ\} with Hurst index 0 &lt; H &lt; 1 and replacing the time parameter with a strictly α-stable Lévy process \{Y(t), t ≥ 0\} in ℝ independent of \{W(t), t ∈ R\}. It is shown that such processes have natural connections to partial differential equations and, when Y is a stable subordinator, can arise as scaling limit of randomly indexed random walks. The existence, joint continuity and sharp Hölder conditions in the set variable of the local times of a d-dimensional α-time fractional Brownian motion X = \{X(t), t ∈ ℝ+\} defined by X(t) = (X1(t), ..., Xd(t)), where t ≥ 0 and X1, ..., Xd are independent copies of Z, are investigated. Our methods rely on the strong local nondeterminism of fractional Brownian motion.},
author = {Nane, Erkan, Wu, Dongsheng, Xiao, Yimin},
journal = {ESAIM: Probability and Statistics},
keywords = {fractional brownian motion; strictlyα-stable Lévy process; α-time brownian motion; α-time fractional brownian motion; partial differential equation; local time; Hölder condition; fractional Brownian motion; strictly -stable Lévy process; -time Brownian motion; -time fractional Brownian motion; PDE},
language = {eng},
pages = {1-24},
publisher = {EDP-Sciences},
title = {α-time fractional brownian motion: PDE connections and local times},
url = {http://eudml.org/doc/273606},
volume = {16},
year = {2012},
}

TY - JOUR
AU - Nane, Erkan
AU - Wu, Dongsheng
AU - Xiao, Yimin
TI - α-time fractional brownian motion: PDE connections and local times
JO - ESAIM: Probability and Statistics
PY - 2012
PB - EDP-Sciences
VL - 16
SP - 1
EP - 24
AB - For 0 &lt; α ≤ 2 and 0 &lt; H &lt; 1, an α-time fractional Brownian motion is an iterated process Z =  {Z(t) = W(Y(t)), t ≥ 0}  obtained by taking a fractional Brownian motion  {W(t), t ∈ ℝ} with Hurst index 0 &lt; H &lt; 1 and replacing the time parameter with a strictly α-stable Lévy process {Y(t), t ≥ 0} in ℝ independent of {W(t), t ∈ R}. It is shown that such processes have natural connections to partial differential equations and, when Y is a stable subordinator, can arise as scaling limit of randomly indexed random walks. The existence, joint continuity and sharp Hölder conditions in the set variable of the local times of a d-dimensional α-time fractional Brownian motion X = {X(t), t ∈ ℝ+} defined by X(t) = (X1(t), ..., Xd(t)), where t ≥ 0 and X1, ..., Xd are independent copies of Z, are investigated. Our methods rely on the strong local nondeterminism of fractional Brownian motion.
LA - eng
KW - fractional brownian motion; strictlyα-stable Lévy process; α-time brownian motion; α-time fractional brownian motion; partial differential equation; local time; Hölder condition; fractional Brownian motion; strictly -stable Lévy process; -time Brownian motion; -time fractional Brownian motion; PDE
UR - http://eudml.org/doc/273606
ER -

References

top
  1. [1] R.J. Adler, The Geometry of Random Fields. Wiley, New York (1981). Zbl0478.60059MR611857
  2. [2] H. Allouba and W. Zheng, Brownian-time processes : the pde connection and the half-derivative generator. Ann. Probab.29 (2001) 1780–1795. Zbl1018.60066MR1880242
  3. [3] F. Aurzada and M. Lifshits, On the Small deviation problem for some iterated processes. Electron. J. Probab.14 (2009) 1992–2010. Zbl1190.60016MR2550290
  4. [4] B. Baeumer, M.M. Meerschaert and E. Nane, Brownian subordinators and fractional Cauchy problems. Trans. Amer. Math. Soc.361 (2009) 3915–3930. Zbl1186.60079MR2491905
  5. [5] B. Baeumer, M.M. Meerschaert and E. Nane, Space-time duality for fractional diffusion. J. Appl. Probab.46 (2009) 1100–1115. Zbl1196.60087MR2582709
  6. [6] L. Beghin, L. Sakhno and E. Orsingher, Equations of Mathematical Physics and composition of Brownian and Cauchy processes. Stoch. Anal. Appl.29 (2011) 551–569. Zbl1223.60083MR2812517
  7. [7] S.M. Berman, Local times and sample function properties of stationary Gaussian processes. Trans. Amer. Math. Soc.137 (1969) 277–299. Zbl0184.40801MR239652
  8. [8] S.M. Berman, Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J.23 (1973) 69–94. Zbl0264.60024MR317397
  9. [9] J. Bertoin, Lévy Processes. Cambridge University Press (1996). Zbl0938.60005MR1406564
  10. [10] K. Burdzy, Some path properties of iterated Brownian motion, in Seminar on Stochastic Processes, edited by E.Çinlar, K.L. Chung and M.J. Sharpe. Birkhäuser, Boston (1993) 67–87. Zbl0789.60060MR1278077
  11. [11] K. Burdzy and D. Khoshnevisan, The level set of iterated Brownian motion, Séminaire de Probabilités XXIX, edited by J. Azéma, M. Emery, P.-A. Meyer and M. Yor. Lect. Notes Math. 1613 (1995) 231–236. Zbl0853.60061MR1459464
  12. [12] K. Burdzy and D. Khoshnevisan, Brownian motion in a Brownian crack. Ann. Appl. Probab.8 (1998) 708–748. Zbl0937.60081MR1627764
  13. [13] E. Csáki, M. Csörgö, A. Földes and P. Révész, The local time of iterated Brownian motion. J. Theoret. Probab.9 (1996) 717–743. Zbl0857.60081MR1400596
  14. [14] J. Cuzick and J. DuPreez, Joint continuity of Gaussian local times. Ann. Probab.10 (1982) 810–817. Zbl0492.60032MR659550
  15. [15] Y. Davydov, The invariance principle for stationary processes. Teor. Verojatnost. i Primenen.15 (1970) 498–509. Zbl0209.48904MR283872
  16. [16] R.D. DeBlassie, Higher order PDE’s and symmetric stable processes. Probab. Theory Relat. Fields129 (2004) 495–536. Zbl1060.60077MR2078980
  17. [17] R.D. DeBlassie, Iterated Brownian motion in an open set. Ann. Appl. Probab.14 (2004) 1529–1558. Zbl1051.60082MR2071433
  18. [18] M. D’Ovidio and E. Orsingher, Composition of processes and related partial differential equations. J. Theor. Probab.24 (2011) 342–375. Zbl1229.60045MR2795043
  19. [19] W. Ehm, Sample function properties of multi-parameter stable processes. Z. Wahrsch. verw. Geb. 56 (1981) 195–228. Zbl0471.60046MR618272
  20. [20] P. Embrechts and M. Maejima, Selfsimilar Processes. Princeton University Press, Princeton (2002). Zbl1008.60003MR1920153
  21. [21] D. Geman and J. Horowitz, Occupation densities. Ann. Probab.8 (1980) 1–67. Zbl0499.60081MR556414
  22. [22] M. Hahn, K. Kobayashi and S. Umarov, Fokker-Plank-Kolmogorv equations associated with SDEs driven by time-changed fractional Brownian motion. Proc. Amer. Math. Soc.139 (2011) 691–705. Zbl1218.60030MR2736349
  23. [23] Y. Hu, Hausdorff and packing measures of the level sets of iterated Brownian motion. J. Theoret. Probab.12 (1999) 313–346. Zbl0935.60066MR1684747
  24. [24] J.P. Kahane, Some Random Series of Functions, 2nd edition. Cambridge University Press (1985). Zbl0805.60007MR833073
  25. [25] D. Khoshnevisan and Y. Xiao, Images of the Brownian sheet. Trans. Amer. Math. Soc.359 (2007) 3125–3151. Zbl1124.60037MR2299449
  26. [26] M.A. Lifshits, Gaussian Random Functions. Kluwer Academic Publishers, Dordrecht (1995). Zbl0832.60002MR1472736
  27. [27] W. Linde and Z. Shi, Evaluating the small deviation probabilities for subordinated Lévy processes. Stoch. Process. Appl.113 (2004) 273–287. Zbl1076.60039MR2087961
  28. [28] E. Nane, Iterated Brownian motion in parabola-shaped domains. Potential Anal.24 (2006) 105–123. Zbl1090.60071MR2217416
  29. [29] E. Nane, Iterated Brownian motion in bounded domains in ℝn. Stoch. Process. Appl.116 (2006) 905–916. Zbl1106.60309MR2254664
  30. [30] E. Nane, Laws of the iterated logarithm for α-time Brownian motion. Electron. J. Probab.11 (2006) 434–459. Zbl1121.60085MR2223043
  31. [31] E. Nane, Higher order PDE’s and iterated processes. Trans. Amer. Math. Soc.360 (2008) 2681–2692. Zbl1157.60071MR2373329
  32. [32] E. Nane, Laws of the iterated logarithm for a class of iterated processes. Statist. Probab. Lett.79 (2009) 1744–1751. Zbl1173.60317MR2566748
  33. [33] E. Orsingher and L. Beghin, Fractional diffusion equations and processes with randomly varying time, Ann. Probab.37 (2009) 206–249. Zbl1173.60027MR2489164
  34. [34] L.D. Pitt, Local times for Gaussian vector fields. Indiana Univ. Math. J.27 (1978) 309–330. Zbl0382.60055MR471055
  35. [35] G. Samorodnitsky and M.S. Taqqu, Stable non-Gaussian Random Processes : Stochastic models with infinite variance. Chapman & Hall, New York (1994). Zbl0925.60027MR1280932
  36. [36] K.I. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press (1999). Zbl1287.60003MR1739520
  37. [37] A.V. Skorokhod, Asymptotic formulas for stable distribution laws. Selected Translations in Mathematical Statistics and Probability 1 (1961) 157–162; Dokl. Akad. Nauk. SSSR 98 (1954) 731–734. Zbl0057.11106MR65839
  38. [38] M. Talagrand, Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Ann. Probab.23 (1995) 767–775. Zbl0830.60034MR1334170
  39. [39] M. Talagrand, Multiple points of trajectories of multiparameter fractional Brownian motion. Probab. Theory Relat. Fields112 (1998) 545–563. Zbl0928.60026MR1664704
  40. [40] M.S. Taqqu, Weak Convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete31 (1975) 287–302. Zbl0303.60033MR400329
  41. [41] S.J. Taylor, Sample path properties of a transient stable process. J. Math. Mech.16 (1967) 1229–1246. Zbl0178.19301MR208684
  42. [42] W. Whitt, Stochastic-Process Limits. Springer, New York (2002). Zbl0993.60001MR1876437
  43. [43] Y. Xiao, Hölder conditions for the local times and Hausdorff measure of the level sets of Gaussian random fields. Probab. Theory Relat. Fields109 (1997) 129–157. Zbl0882.60035MR1469923
  44. [44] Y. Xiao, Local times and related properties of multi-dimensional iterated Brownian motion. J. Theoret. Probab.11 (1998) 383–408. Zbl0914.60063MR1622577

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