Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Optimal transportation for multifractal random measures and applications

Rémi Rhodes and Vincent Vargas

Full-text: Open access

Abstract

In this paper, we study optimal transportation problems for multifractal random measures. Since these measures are much less regular than optimal transportation theory requires, we introduce a new notion of transportation which is intuitively some kind of multistep transportation. Applications are given for construction of multifractal random changes of times and to the existence of random metrics, the volume forms of which coincide with the multifractal random measures.

Résumé

Dans ce papier, nous étudions des problèmes de transport optimal pour des mesures aléatoires multifractales. Puisque ces mesures sont beaucoup moins régulières que ce que la théorie requiert habituellement, nous introduisons une nouvelle notion de transport qui peut être vue intuitivement comme du transport à étapes multiples. En application, nous construisons des changements de temps multifractals et nous établissons l’existence de métriques aléatoires pour lesquelles les formes volume sont des mesures aléatoires multifractales.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 1 (2013), 119-137.

Dates
First available in Project Euclid: 29 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1359470128

Digital Object Identifier
doi:10.1214/11-AIHP443

Mathematical Reviews number (MathSciNet)
MR3060150

Zentralblatt MATH identifier
1296.60130

Subjects
Primary: 60G57: Random measures 49J55: Problems involving randomness [See also 93E20] 28A80: Fractals [See also 37Fxx]
Secondary: 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]

Keywords
Random measures Multifractal processes Optimal transportation Random metric

Citation

Rhodes, Rémi; Vargas, Vincent. Optimal transportation for multifractal random measures and applications. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 1, 119--137. doi:10.1214/11-AIHP443. https://projecteuclid.org/euclid.aihp/1359470128


Export citation

References

  • [1] E. Bacry and J. F. Muzy. Log-infinitely divisible multifractal processes. Comm. Math. Phys. 236 (2003) 449–475.
  • [2] I. Benjamini and O. Schramm. KPZ in one dimensional random geometry of multiplicative cascades. Comm. Math. Phys. 289 (2009) 653–662.
  • [3] B. Duplantier and S. Sheffield. Liouville quantum gravity and KPZ. Invent. Math. 185 (2011) 333–393.
  • [4] A. H. Fan. Sur le chaos de Lévy d’indice $0<\alpha<1$. Ann. Sci. Math. Québec 21 (1997) 53–66.
  • [5] J.-P. Kahane. Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9 (1985) 105–150.
  • [6] R. Rhodes and V. Vargas. KPZ formula for log-infinitely divisible multifractal random measures. ESAIM Probab. Stat. 15 (2011) 358–371.
  • [7] R. Rhodes and V. Vargas. Multidimensional multifractal random measures. Electron. J. Probab. 15 (2010) 241–258.
  • [8] C. Villani. Optimal Transport, Old and New. Grundlehren Math. Wiss. 338. Springer, Berlin.
  • [9] C. Villani. Topics in Optimal Transportations. Grad. Stud. Math. 58. Amer. Math. Soc., Providence, RI, 2003.
  • [10] E. C. Waymire and S. C. William. Multiplicative cascades: Dimension spectra and dependence. J. Fourier Anal. Appl. (Special Issue: Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993)) (1995) 589–609.