Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A uniform dimension result for two-dimensional fractional multiplicative processes

Xiong Jin

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Abstract

Given a two-dimensional fractional multiplicative process $(F_{t})_{t\in[0,1]}$ determined by two Hurst exponents $H_{1}$ and $H_{2}$, we show that there is an associated uniform Hausdorff dimension result for the images of subsets of $[0,1]$ by $F$ if and only if $H_{1}=H_{2}$.

Résumé

Etant donné un processus multiplicatif fractionnaire bi-dimensionnel $(F_{t})_{t\in[0,1]}$ déterminé par deux exposants de Hurst $H_{1}$ et $H_{2}$, nous montrons l’existence d’un résultat uniforme pour la dimension de Hausdorff des images des sous-ensembles de $[0,1]$ par $F$ si et seulement si $H_{1}=H_{2}$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 2 (2014), 512-523.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1395856138

Digital Object Identifier
doi:10.1214/12-AIHP509

Mathematical Reviews number (MathSciNet)
MR3189082

Zentralblatt MATH identifier
1292.60049

Subjects
Primary: 60G18: Self-similar processes
Secondary: 28A78: Hausdorff and packing measures

Keywords
Hausdorff dimension Fractional multiplicative processes Uniform dimension result Level sets

Citation

Jin, Xiong. A uniform dimension result for two-dimensional fractional multiplicative processes. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 2, 512--523. doi:10.1214/12-AIHP509. https://projecteuclid.org/euclid.aihp/1395856138


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References

  • [1] J. Barral and X. Jin. Multifractal analysis of complex random cascades. Comm. Math. Phys. 297 (2010) 129–168.
  • [2] J. Barral, X. Jin and B. Mandelbrot. Convergence of complex multiplicative cascades. Ann. Appl. Probab. 20 (2010) 1219–1252.
  • [3] J. Barral and B. Mandelbrot. Fractional multiplicative processes. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 1116–1129.
  • [4] I. Benjamini and O. Schramm. KPZ in one dimensional random geometry of multiplicative cascades. Comm. Math. Phys. 289 (2009) 653–662.
  • [5] R. M. Blumenthal and R. K. Getoor. A dimension theorem for sample functions of stable processes. Illinois J. Math. 4 (1960) 370–375.
  • [6] R. M. Blumenthal and R. K. Getoor. Sample functions of stochastic processes with stationary independent increments. J. Math. Mech. 10 (1961) 493–516.
  • [7] B. Duplantier and S. Sheffield. Liouville quantum gravity and KPZ. Invent. Math. 185 (2011) 333–393.
  • [8] K. Falconer. Fractal Geometry: Mathematical Foundations and Applications, 2nd edition. Wiley, Hoboken, NJ, 2003.
  • [9] J. Hawkes. Some dimension theorems for the sample functions of stable processes. Indiana Univ. Math. J. 20 (1970/71) 733–738.
  • [10] J. Hawkes and W. E. Pruitt. Uniform dimension results for processes with independent increments. Z. Wahrsch. Verw. Gebiete 28 (1973/74) 277–288.
  • [11] X. Jin. The graph and range singularity spectra of $b$-adic independent cascade functions. Adv. Math. 226 (2011) 4987–5017.
  • [12] X. Jin. Dimension result and KPZ formula for two-dimensional multiplicative cascade processes. Ann. Probab. 40 (2012) 1–18.
  • [13] J.-P. Kahane. Some Random Series of Functions, 2nd edition. Cambridge Studies in Advanced Mathematics 5. Cambridge Univ. Press, Cambridge, 1985.
  • [14] J.-P. Kahane and J. Peyrière. Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22 (1976) 131–145.
  • [15] R. Kaufman. Une propriété métrique du mouvement brownien. C. R. Acad. Sci. Paris Sér. A-B 268 (1969) A727–A728.
  • [16] D. Khoshnevisan and Y. Xiao. Lévy processes: capacity and Hausdorff dimension. Ann. Probab. 33 (2005) 841–878.
  • [17] P. Lévy. La mesure de Hausdorff de la courbe du mouvement brownien. Giorn. Ist. Ital. Attuari 16 (1953) 1–37.
  • [18] H. P. McKean, Jr. Hausdorff-Besicovitch dimension of Brownian motion paths. Duke Math. J. 22 (1955) 229–234.
  • [19] P. W. Millar. Path behavior of processes with stationary independent increments. Z. Wahrsch. Verw. Gebiete 17 (1971) 53–73.
  • [20] R. Rhodes and V. Vargas. KPZ formula for log-infinitely divisible multifractal random measures. ESAIM: Probab. Stat. 15 (2011) 358–371.
  • [21] S. J. Taylor. The Hausdorff $\alpha$-dimensional measure of Brownian paths in $n$-space. Math. Proc. Cambridge Philos. Soc. 49 (1953) 31–39.
  • [22] S. J. Taylor. The measure theory of random fractals. Math. Proc. Cambridge Philos. Soc. 100 (1986) 383–406.
  • [23] D. Wu and Y. Xiao. Uniform dimension results for Gaussian random fields. Sci. China Ser. A 52 (2009) 1478–1496.
  • [24] Y. Xiao. Dimension results for Gaussian vector fields and index-$\alpha$ stable fields. Ann. Probab. 23 (1995) 273–291.
  • [25] Y. Xiao. Random fractals and Markov processes. In Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2 261–338. Proc. Sympos. Pure Math. 72. Amer. Math. Soc., Providence, RI, 2004.