Electronic Journal of Probability

Multifractal analysis for the occupation measure of stable-like processes

Stéphane Seuret and Xiaochuan Yang

Full-text: Open access

Abstract

In this article, we investigate the local behavior of the occupation measure $\mu $ of a class of real-valued Markov processes $\mathcal{M} $, defined via a SDE. This (random) measure describes the time spent in each set $A\subset \mathbb{R} $ by the sample paths of $\mathcal{M} $. We compute the multifractal spectrum of $\mu $, which turns out to be random, depending on the trajectory. This remarkable property is in sharp contrast with the results previously obtained on occupation measures of other processes (such as Lévy processes), where the multifractal spectrum is usually deterministic, almost surely. In addition, the shape of this multifractal spectrum is very original, reflecting the richness and variety of the local behavior. The proof is based on new methods, which lead for instance to fine estimates on Hausdorff dimensions of certain jump configurations in Poisson point processes.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 47, 36 pp.

Dates
Received: 23 June 2016
Accepted: 9 March 2017
First available in Project Euclid: 30 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1496109646

Digital Object Identifier
doi:10.1214/17-EJP48

Mathematical Reviews number (MathSciNet)
MR3661661

Zentralblatt MATH identifier
1364.60075

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J25: Continuous-time Markov processes on general state spaces 60J75: Jump processes 28A80: Fractals [See also 37Fxx] 28A78: Hausdorff and packing measures

Keywords
Markov and Lévy processes occupation measure Hausdorff measure and dimension

Rights
Creative Commons Attribution 4.0 International License.

Citation

Seuret, Stéphane; Yang, Xiaochuan. Multifractal analysis for the occupation measure of stable-like processes. Electron. J. Probab. 22 (2017), paper no. 47, 36 pp. doi:10.1214/17-EJP48. https://projecteuclid.org/euclid.ejp/1496109646


Export citation

References

  • [1] David Applebaum, Lévy processes and stochastic calculus, second ed., Cambridge Studies in Advanced Mathematics, vol. 116, Cambridge University Press, Cambridge, 2009.
  • [2] Paul Balança, Fine regularity of Lévy processes and linear (multi)fractional stable motion, Electron. J. Probab. 19 (2014), no. 101, 37.
  • [3] Paul Balança, Uniform multifractal structure of stable trees, Arxiv (2015).
  • [4] Julien Barral, Nicolas Fournier, Stéphane Jaffard, and Stéphane Seuret, A pure jump Markov process with a random singularity spectrum, Ann. Probab. 38 (2010), no. 5, 1924–1946.
  • [5] Julien Barral and Stéphane Seuret, The singularity spectrum of Lévy processes in multifractal time, Adv. Math. 214 (2007), no. 1, 437–468.
  • [6] Richard F. Bass, Uniqueness in law for pure jump Markov processes, Probab. Theory Related Fields 79 (1988), no. 2, 271–287.
  • [7] Julien Berestycki, Nathanaël Berestycki, and Jason Schweinsberg, Beta-coalescents and continuous stable random trees, Ann. Probab. 35 (2007), no. 5, 1835–1887.
  • [8] Jean Bertoin, Lévy processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996.
  • [9] R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc. 95 (1960), 263–273.
  • [10] Amir Dembo, Yuval Peres, Jay Rosen, and Ofer Zeitouni, Thick points for spatial Brownian motion: multifractal analysis of occupation measure, Ann. Probab. 28 (2000), no. 1, 1–35.
  • [11] Arnaud Durand, Singularity sets of Lévy processes, Probab. Theory Related Fields 143 (2009), no. 3-4, 517–544.
  • [12] Arnaud Durand and Stéphane Jaffard, Multifractal analysis of Lévy fields, Probab. Theory Related Fields 153 (2012), no. 1-2, 45–96.
  • [13] Kenneth Falconer, Fractal geometry, second ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003, Mathematical foundations and applications.
  • [14] Donald Geman and Joseph Horowitz, Occupation densities, Ann. Probab. 8 (1980), no. 1, 1–67.
  • [15] Xiaoyu Hu and S. James Taylor, The multifractal structure of stable occupation measure, Stochastic Process. Appl. 66 (1997), no. 2, 283–299.
  • [16] Xiaoyu Hu and S. James Taylor, Multifractal structure of a general subordinator, Stochastic Process. Appl. 88 (2000), no. 2, 245–258.
  • [17] Stéphane Jaffard, The multifractal nature of Lévy processes, Probab. Theory Related Fields 114 (1999), no. 2, 207–227.
  • [18] Stéphane Jaffard, Wavelet techniques in multifractal analysis, Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2, Proc. Sympos. Pure Math., vol. 72, Amer. Math. Soc., Providence, RI, 2004, pp. 91–151.
  • [19] Davar Khoshnevisan, Kunwoo Kim, and Yimin Xiao, Intermittency and multifractality: A case study via parabolic stochastic pdes, Ann. Probab., in press.
  • [20] Laurence Marsalle, Slow points and fast points of local times, Ann. Probab. 27 (1999), no. 1, 150–165.
  • [21] Peter Mörters and Narn-Rueih Shieh, Thin and thick points for branching measure on a Galton-Watson tree, Statist. Probab. Lett. 58 (2002), no. 1, 13–22.
  • [22] Peter Mörters and Narn-Rueih Shieh, On the multifractal spectrum of the branching measure on a Galton-Watson tree, J. Appl. Probab. 41 (2004), no. 4, 1223–1229.
  • [23] Leonid Mytnik and Vitali Wachtel, Multifractal analysis of superprocesses with stable branching in dimension one, Ann. Probab. 43 (2015), no. 5, 2763–2809.
  • [24] Edwin A. Perkins and S. James Taylor, The multifractal structure of super-Brownian motion, Ann. Inst. H. Poincaré Probab. Statist. 34 (1998), no. 1, 97–138.
  • [25] Narn-Rueih Shieh and S. James Taylor, Logarithmic multifractal spectrum of stable occupation measure, Stochastic Process. Appl. 75 (1998), no. 2, 249–261.
  • [26] Liping Xu, The multifractal nature of boltzmann processes, Stochastic Process. Appl. 126 (April 2015), no. 8. DOI: 10.1016/j.spa.2016.01.008.
  • [27] Xiaochuan Yang, Hausdorff dimension of the range and the graph of stable-like processes, Arxiv, e-print (2015).
  • [28] Xiaochuan Yang, Multifractality of jump diffusion processes, Arxiv, e-print (2015).