Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T01:35:48.808Z Has data issue: false hasContentIssue false
  • English
  • Français

α-time fractional Brownian motion: PDEconnections and local times∗

Published online by Cambridge University Press:  09 March 2012

Erkan Nane ,
Dongsheng Wu  and
Yimin Xiao
Erkan Nane
Affiliation:
Department of Mathematics and Statistics, Auburn University, 221 Parker Hall, Auburn, AL 36849, USA. www.duc.auburn.edu/˜ezn0001/. nane@stt.msu.edu
Dongsheng Wu
Affiliation:
Department of Mathematical Sciences, 201J Shelby Center, University of Alabama in Huntsville, Huntsville, AL 35899, USA; http://webpages.uah.edu/˜dw0001. dongsheng.wu@uah.edu
Yimin Xiao
Affiliation:
Department of Statistics and Probability, A-413 Wells Hall, Michigan State University, East Lansing, MI 48824, USA; http://www.stt.msu.edu/˜xiaoyimi. xiao@stt.msu.edu
Get access

Abstract

For 0 < α ≤ 2 and 0 < H < 1, anα-time fractional Brownian motion is an iterated processZ =  {Z(t) = W(Y(t)), t ≥ 0} obtained by taking a fractional Brownian motion {W(t), t ∈ ℝ} with Hurst index0 < H < 1 and replacing the time parameter with astrictly α-stable Lévy process {Y(t), t ≥ 0} in ℝ independent of {W(t), t ∈ R}. It is shown that suchprocesses have natural connections to partial differential equations and, whenY is a stable subordinator, can arise as scaling limit of randomlyindexed random walks. The existence, joint continuity and sharp Hölder conditions in theset variable of the local times of a d-dimensionalα-time fractional Brownian motionX = {X(t), t ∈ ℝ+} defined by X(t) = (X1(t), ...,Xd(t)), where t ≥ 0 andX1, ..., Xdare independent copies of Z, are investigated. Our methods rely on thestrong local nondeterminism of fractional Brownian motion.

Keywords

Fractional Brownian motionstrictlyα-stable Lévy processα-time Brownian motionα-time fractional Brownian motionpartial differential equationlocal timeHölder condition.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 1 - 24
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.J. Adler, The Geometry of Random Fields. Wiley, New York (1981).
Allouba, H. and Zheng, W., Brownian-time processes : the pde connection and the half-derivative generator. Ann. Probab. 29 (2001) 1780–1795. Google Scholar
Aurzada, F. and Lifshits, M., On the Small deviation problem for some iterated processes. Electron. J. Probab. 14 (2009) 1992–2010. Google Scholar
Baeumer, B., Meerschaert, M.M. and Nane, E., Brownian subordinators and fractional Cauchy problems. Trans. Amer. Math. Soc. 361 (2009) 3915–3930. Google Scholar
Baeumer, B., Meerschaert, M.M. and Nane, E., Space-time duality for fractional diffusion. J. Appl. Probab. 46 (2009) 1100–1115. Google Scholar
Beghin, L., Sakhno, L. and Orsingher, E., Equations of Mathematical Physics and composition of Brownian and Cauchy processes. Stoch. Anal. Appl. 29 (2011) 551–569. Google Scholar
Berman, S.M., Local times and sample function properties of stationary Gaussian processes. Trans. Amer. Math. Soc. 137 (1969) 277–299. Google Scholar
Berman, S.M., Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23 (1973) 69–94. Google Scholar
J. Bertoin, Lévy Processes. Cambridge University Press (1996).
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.
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.
Burdzy, K. and Khoshnevisan, D., Brownian motion in a Brownian crack. Ann. Appl. Probab. 8 (1998) 708–748. Google Scholar
Csáki, E., Csörgö, M., Földes, A. and Révész, P., The local time of iterated Brownian motion. J. Theoret. Probab. 9 (1996) 717–743. Google Scholar
Cuzick, J. and DuPreez, J., Joint continuity of Gaussian local times. Ann. Probab. 10 (1982) 810–817. Google Scholar
Davydov, Y., The invariance principle for stationary processes. Teor. Verojatnost. i Primenen. 15 (1970) 498–509. Google Scholar
DeBlassie, R.D., Higher order PDE’s and symmetric stable processes. Probab. Theory Relat. Fields 129 (2004) 495–536. Google Scholar
DeBlassie, R.D., Iterated Brownian motion in an open set. Ann. Appl. Probab. 14 (2004) 1529–1558. Google Scholar
D’Ovidio, M. and Orsingher, E., Composition of processes and related partial differential equations. J. Theor. Probab. 24 (2011) 342–375. Google Scholar
Ehm, W., Sample function properties of multi-parameter stable processes. Z. Wahrsch. verw. Geb. 56 (1981) 195–228. Google Scholar
P. Embrechts and M. Maejima, Selfsimilar Processes. Princeton University Press, Princeton (2002).
Geman, D. and Horowitz, J., Occupation densities. Ann. Probab. 8 (1980) 1–67. Google Scholar
Hahn, M., Kobayashi, K. and Umarov, S., Fokker-Plank-Kolmogorv equations associated with SDEs driven by time-changed fractional Brownian motion. Proc. Amer. Math. Soc. 139 (2011) 691–705. Google Scholar
Hu, Y., Hausdorff and packing measures of the level sets of iterated Brownian motion. J. Theoret. Probab. 12 (1999) 313–346. Google Scholar
J.P. Kahane, Some Random Series of Functions, 2nd edition. Cambridge University Press (1985).
Khoshnevisan, D. and Xiao, Y., Images of the Brownian sheet. Trans. Amer. Math. Soc. 359 (2007) 3125–3151. Google Scholar
M.A. Lifshits, Gaussian Random Functions. Kluwer Academic Publishers, Dordrecht (1995).
Linde, W. and Shi, Z., Evaluating the small deviation probabilities for subordinated Lévy processes. Stoch. Process. Appl. 113 (2004) 273–287. Google Scholar
Nane, E., Iterated Brownian motion in parabola-shaped domains. Potential Anal. 24 (2006) 105–123. Google Scholar
Nane, E., Iterated Brownian motion in bounded domains in ℝn. Stoch. Process. Appl. 116 (2006) 905–916. Google Scholar
Nane, E., Laws of the iterated logarithm for α-time Brownian motion. Electron. J. Probab. 11 (2006) 434–459. Google Scholar
Nane, E., Higher order PDE’s and iterated processes. Trans. Amer. Math. Soc. 360 (2008) 2681–2692. Google Scholar
Nane, E., Laws of the iterated logarithm for a class of iterated processes. Statist. Probab. Lett. 79 (2009) 1744–1751. Google Scholar
Orsingher, E. and Beghin, L., Fractional diffusion equations and processes with randomly varying time, Ann. Probab. 37 (2009) 206–249. Google Scholar
Pitt, L.D., Local times for Gaussian vector fields. Indiana Univ. Math. J. 27 (1978) 309–330. Google Scholar
G. Samorodnitsky and M.S. Taqqu, Stable non-Gaussian Random Processes : Stochastic models with infinite variance. Chapman & Hall, New York (1994).
K.I. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press (1999).
Skorokhod, A.V., 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. Google Scholar
Talagrand, M., Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Ann. Probab. 23 (1995) 767–775. Google Scholar
Talagrand, M., Multiple points of trajectories of multiparameter fractional Brownian motion. Probab. Theory Relat. Fields 112 (1998) 545–563. Google Scholar
Taqqu, M.S., Weak Convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 (1975) 287–302. Google Scholar
Taylor, S.J., Sample path properties of a transient stable process. J. Math. Mech. 16 (1967) 1229–1246. Google Scholar
W. Whitt, Stochastic-Process Limits. Springer, New York (2002).
Xiao, Y., Hölder conditions for the local times and Hausdorff measure of the level sets of Gaussian random fields. Probab. Theory Relat. Fields 109 (1997) 129–157. Google Scholar
Xiao, Y., Local times and related properties of multi-dimensional iterated Brownian motion. J. Theoret. Probab. 11 (1998) 383–408. Google Scholar