Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T20:34:34.141Z Has data issue: false hasContentIssue false
  • English
  • Français

Simultaneous Multifractal Analysis of the Branching and Visibility Measure on a Galton-Watson Tree

Part of: Classical measure theory Markov processes

Published online by Cambridge University Press:  01 July 2016

Adam L. Kinnison  and
Peter Mörters
Adam L. Kinnison*
Affiliation:
University of Bath
Peter Mörters*
Affiliation:
University of Bath
*
∗ Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.
∗ Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

On the boundary of a Galton-Watson tree we can define the visibility measure by splitting mass equally between the children of each vertex, and the branching measure by splitting unit mass equally between all vertices in the nth generation and then letting n go to infinity. The multifractal structure of each of these measures is well studied. In this paper we address the question of a joint multifractal spectrum, i.e. we ask for the Hausdorff dimension of the boundary points which simultaneously have an unusual local dimension for both these measures. The resulting two-parameter spectrum exhibits a number of surprising new features, among them the emergence of a swallowtail-shaped spectrum for the visibility measure in the presence of a nontrivial condition on the branching measure.

Keywords

Multifractal spectrumtwo-parameter spectrummixed spectrumtwo-dimensional multifractal analysisrandom treeself-similar fractalbranching processpercolation

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 28A80: Fractals
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 1 , March 2010 , pp. 226 - 245
Copyright
Copyright © Applied Probability Trust 2010 

References

Anh, V. V., Leonenko, N. N. and Shieh, N.-R. (2008). Multifractality of products of geometric Ornstein–Uhlenbeck-type processes. Adv. Appl. Prob. 40, 1129–1156.Google Scholar
Arbeiter, M. and Patzschke, N. (1996). Random self-similar multifractals. Math. Nachr. 181, 5–42.Google Scholar
Barreira, L. and Saussol, B. (2001). Variational principles and mixed multifractal spectra. Trans. Amer. Math. Soc. 353, 3919–3944.Google Scholar
Barreira, L., Saussol, B. and Schmeling, J. (2002). Higher-dimensional multifractal analysis. J. Math. Pures Appl. 81, 67–91.CrossRefGoogle Scholar
Benzi, R., Paladin, G., Parisi, G. and Vulpiani, A. (1984). On the multifractal nature of fully developed turbulence and chaotic systems. J. Phys. A 17, 3521–3531.Google Scholar
Berestycki, J. (2003). Multifractal spectra of fragmentation processes. J. Statist. Phys. 113, 411–430.Google Scholar
Biggins, J. D. and Bingham, N. H. (1993). Large deviations in the supercritical branching process. Adv. Appl. Prob. 25, 757–772.Google Scholar
Brown, G., Michon, G. and Peyriere, J. (1992). On the multifractal analysis of measures. J. Statist. Phys. 66, 775–790.Google Scholar
Cawley, R. and Mauldin, R. D. (1992). Multifractal decompositions of Moran fractals. Adv. Math. 92, 196–236.Google Scholar
Dubuc, M. S. (1971). La densité de la loi-limite d'un processus en cascade expansif. Z. Wahrscheinlichkeitsth. 19, 281–290.CrossRefGoogle Scholar
Falconer, K. (2003). Fractal Geometry, 2nd edn. John Wiley, Hoboken, NJ.Google Scholar
Fleischmann, K. and Wachtel, V. (2007). Lower deviation probabilities for supercritical Galton–Watson processes. Ann. Inst. H. Poincaré Prob. Statist. 43, 233–255.Google Scholar
Frisch, U. and Parisi, G. (1985). On the singularity structure of fully developed turbulence. In Proc. Int. Sum. School Enrico Fermi, North Holland, Amsterdam, pp. 84–88.Google Scholar
Halsey, T. C. et al. (1986). Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. A 33, 1141–1151.Google Scholar
Hawkes, J. (1981). Trees generated by a simple branching process. J. London Math. Soc. 24, 373–384.Google Scholar
Kinnison, A. L. (2008). The multifractal spectrum of harmonic measure for forward moving random walks on a Galton–Watson tree. Statist. Prob. Lett. 78, 3114–3121.Google Scholar
Klenke, A. and Mörters, P. (2005). The multifractal spectrum of Brownian intersection local times. Ann. Prob. 33, 1255–1301.Google Scholar
Lau, K.-S. and Ngai, S.-M. (1999). Multifractal measures and a weak separation condition. Adv. Math. 141, 45–96.Google Scholar
Liu, Q. (2001). Local dimensions of the branching measure on a Galton–Watson tree. Ann. Inst. H. Poincaré Prob. Statist. 37, 195–222.Google Scholar
Liu, Q. and Rouault, A. (1997). On two measures defined on the boundary of a branching tree. In Classical and Modern Branching Processes (Minneapolis, MN, 1994; IMA Vol. Math. Appl. 84), Springer, New York, pp. 187–201.Google Scholar
Lyons, R. (1990). Random walks and percolation on trees. Ann. Prob. 18, 931–958.Google Scholar
Lyons, R. and Peres, Y. (2010). Probability on Trees and Networks. In preparation. Available at http://php. indiana.edu/∼rdlyons/prbtree/prbtree.html.Google Scholar
Lyons, R., Pemantle, R. and Peres, Y. (1995). Ergodic theory on Galton–Watson trees: speed of random walk and dimension of harmonic measure. Ergodic Theory Dynam. Syst. 15, 593–619.Google Scholar
Mandelbrot, B. B. (1974). Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331–358.Google Scholar
Mannersalo, P., Norros, I. and Riedi, R. H. (2002). Multifractal products of stochastic processes: construction and some basic properties. Adv. Appl. Prob. 34, 888–903.Google Scholar
Mörters, P. (2009). Why study multifractal spectra? In Trends in Stochastic Analysis (London Math. Soc. Lecture Notes Ser. 353), Cambridge University Press, pp. 99–120.Google Scholar
Mörters, P. and Ortgiese, M. (2008). Small value probabilities via the branching tree heuristic. Bernoulli 14, 277–299.Google Scholar
Mörters, P. and Shieh, N.-R. (2004). On the multifractal spectrum of the branching measure of a Galton–Watson tree. J. Appl. Prob. 41, 1223–1229.Google Scholar
Olsen, L. (1995). A multifractal formalism. Adv. Math. 116, 82–196.Google Scholar
Olsen, L. (2003). Mixed divergence points of self-similar measures. Indiana Univ. Math. J. 52, 1343–1372.Google Scholar
Olsen, L. (2003). Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. J. Math. Pures Appl. 82, 1591–1649.Google Scholar
Olsen, L. (2005). Mixed generalized dimensions of self-similar measures. J. Math. Anal. Appl. 306, 516–539.CrossRefGoogle Scholar
Olsen, L. and Winter, S. (2003). Normal and non-normal points of self-similar sets and divergence points of self-similar measures. J. London Math. Soc. 67, 103–122.Google Scholar
Peres, Y. (1999). Probability on trees: an introductory climb. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997; Lecture Notes Math. 1717), Springer, Berlin, pp. 193–280.Google Scholar
Perkins, E. A. and Taylor, S. J. (1998). The multifractal structure of super-Brownian motion. Ann. Inst. H. Poincaré Prob. Statist. 34, 97–138.Google Scholar
Rand, D. A. (1989). The singularity spectrum f(a) for cookie-cutters. Ergodic Theory Dynam. Systems 9, 527–541.Google Scholar
Shieh, N.-R. and Taylor, S. J. (2002). Multifractal spectra of branching measure on a Galton–Watson tree. J. Appl. Prob. 39, 100–111.Google Scholar