Skip to main content
SpringerLink
Log in

Trigonometric multiplicative chaos and applications to random distributions

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

The random trigonometric series \(\sum\nolimits_{n = 1}^\infty {{\rho _n}\cos \left( {nt + {\omega _n}} \right)} \) on the circle \(\mathbb{T}\) are studied under the conditions ∑∣ρn∣2 = ∞ and ρn → 0, where {ωn} are independent and uniformly distributed random variables on \(\mathbb{T}\). They are almost surely not Fourier-Stieltjes series but determine pseudo-functions. This leads us to develop the theory of trigonometric multiplicative chaos, which produces a class of random measures. The kernel and the image of chaotic operators are fully studied and the dimensions of chaotic measures are exactly computed. The behavior of the partial sums of the above series is proved to be multifractal. Our theory holds on the torus \({\mathbb{T}^d}\) of dimension d ⩾ 1.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bacry E, Muzy J F. Log-infinitely divisible multifractal processes. Comm Math Phys, 2003, 236: 449–475

    Article  MathSciNet  MATH  Google Scholar 

  2. Barral J. Moments, continuite, et analyse multifractale des martingales de Mandelbrot. Probab Theory Related Fields, 1999, 113: 535–569

    Article  MathSciNet  MATH  Google Scholar 

  3. Barral J. Generalized vector multiplicative cascades. Adv in Appl Probab, 2001, 33: 874–895

    Article  MathSciNet  MATH  Google Scholar 

  4. Barral J, Fan A H. Covering numbers of different points in Dvoretzky covering. Bull Sci Math, 2005, 129: 275–317

    Article  MathSciNet  MATH  Google Scholar 

  5. Barral J, Fan A H, Peyrière J. Mesures engendrées par multiplications. In: Quelques Interactions Entre Analyse, Probabilités et Fractals. Panoramas et Synthesès, vol. 32. Paris: Soc Math France, 2010, 57–189

    MATH  Google Scholar 

  6. Barral J, Jin X, Rhodes R, et al. Gaussian multiplicative chaos and KPZ duality. Comm Math Phys, 2013, 323: 451–485

    Article  MathSciNet  MATH  Google Scholar 

  7. Barral J, Kupiainen A, Nikula M, et al. Basic properties of critical lognormal multiplicative chaos. Ann Probab, 2015, 43: 2205–2249

    Article  MathSciNet  MATH  Google Scholar 

  8. Bary N. A Treatise on Trigonometric Series, Volume 1. Oxford: Pergamon Press, 1964

    Google Scholar 

  9. Ben Nasr F. Mesures alátoires de Mandelbrot associées à des substitutions. C R Acad Sci Paris Sér I Math, 1987, 304: 255–258

    MathSciNet  MATH  Google Scholar 

  10. Benjamini I, Schramm O. KPZ in one dimensional random geometry of multiplicative cascades. Comm Math Phys, 2009, 289: 653–662

    Article  MathSciNet  MATH  Google Scholar 

  11. Billard P. Series de Fourier aléatoirement bornées, continues, uniformément convergentes. Ann Sci Éc Norm Supér (4), 1965, 82: 131–179

    Article  MATH  Google Scholar 

  12. Duplantier B, Sheffield S. Liouville quantum gravity and KPZ. Invent Math, 2011, 185: 333–393

    Article  MathSciNet  MATH  Google Scholar 

  13. Ellis R S. Entropy, Large Deviations, and Statistical Mechanics. New York: Springer-Verlag, 1985

    Book  MATH  Google Scholar 

  14. Falconer K, Jin X. Exact dimensionality and projection properties of Gaussian multiplicative chaos measures. Trans Amer Math Soc, 2019, 372: 2921–2957

    Article  MathSciNet  MATH  Google Scholar 

  15. Fan A H. Chaos additifs et chaos multiplicatifs de Lévy. C R Acad Sci Paris Sér I, 1989, 308: 151–154

    MathSciNet  Google Scholar 

  16. Fan A H. Recouvrement aléatoire et décompositions de mesures. Paris: Publication d’Orsay, 1989

    Google Scholar 

  17. Fan A H. Sur quelques processus de Naissance et de mort. C R Acad Sci Paris Sér I, 1990, 310: 441–444

    MathSciNet  MATH  Google Scholar 

  18. Fan A H. Equivalence et orthogonalité des mesures aléatoires engendrées par martingales positives homogènes. Studia Math, 1991, 98: 249–266

    Article  MathSciNet  MATH  Google Scholar 

  19. Fan A H. Quelques propriétés de produits de Riesz. Bull Sci Math, 1993, 117: 421–439

    MathSciNet  MATH  Google Scholar 

  20. Fan A H. Sur les dimensions de mesures. Studia Math, 1994, 111: 1–17

    Article  MathSciNet  MATH  Google Scholar 

  21. Fan A H. Recouvrements par des intervalles de longueurs alátoires. Ann Sc Math Québec, 1995, 19: 49–64

    MATH  Google Scholar 

  22. Fan A H. Sur la Lp convergence des martingales liés au recouvrement. J Appl Probab, 1995, 32: 668–678

    Article  MathSciNet  MATH  Google Scholar 

  23. Fan A H. Sur le chaos de Lévy d’indice 0 < α < 1. Ann Sc Math Québec, 1997, 21: 53–66

    Google Scholar 

  24. Fan A H. How many intervals cover a point in Dvoretzky covering? Israel J Math, 2002, 131: 157–184

    Article  MathSciNet  MATH  Google Scholar 

  25. Fan A H. On Markov-Mandelbrot martingales. J Math Pures Appl (9), 2002, 81: 967–982

    Article  MathSciNet  MATH  Google Scholar 

  26. Fan A H. Limsup deviations on trees. Anal Theory Appl, 2004, 20: 113–148

    Article  MathSciNet  MATH  Google Scholar 

  27. Fan A H, Kahane J P. Rareté des intervalles recouvrant un point dans un recouvrement aléatoire. Ann Inst Henri Poincaré Probab Stat, 1993, 29: 453–466

    MATH  Google Scholar 

  28. Fan A H, Kahane J P. How many intervals cover a point in random dyadic covering? Port Math, 2001, 58: 59–75

    MathSciNet  MATH  Google Scholar 

  29. Fan A H, Kahane J P. Decomposition principle and random cascades. In: Third International Congress of Chinese Mathematicians. AMS/IP Studies in Advanced Mathematics, vol. 42. Providence: Amer Math Soc, 2008, 447–456

    MATH  Google Scholar 

  30. Fan A H, Karagulyan D. On μ-Dvoretzky random covering of the circle. Bernoulli, 2021, 27: 1270–1290

    Article  MathSciNet  MATH  Google Scholar 

  31. Gwynne E, Miller J. Existence and uniqueness of the Liouville quantum gravity metric for γ ∈ (0, 2). Invent Math, 2021, 223: 213–333

    Article  MathSciNet  MATH  Google Scholar 

  32. Junnila J. On the multiplicative chaos of non-Gaussian log-correlated fields. Int Math Res Not IMRN, 2020, 19: 6169–6196

    Article  MathSciNet  MATH  Google Scholar 

  33. Kahane J-P. Propriétés locales des fonctions à séries de Fourier aléatoires. Studia Math, 1960, 19: 1–25

    Article  MathSciNet  MATH  Google Scholar 

  34. Kahane J-P. Sur le chaos multiplicatif. Ann Sc Math Québec, 1985, 9: 105–150

    MathSciNet  MATH  Google Scholar 

  35. Kahane J-P. Some Random Series of Functions, 2nd ed. Cambridge Studies in Advanced Mathematics, vol. 5. Cambridge: Cambridge University Press, 1985

    MATH  Google Scholar 

  36. Kahane J-P. Intervalles aléatoires et décomposition des mesures. C R Acad Sci Paris Sér I, 1987, 304: 551–554

    MathSciNet  MATH  Google Scholar 

  37. Kahane J-P. Positive martingales and random measures. Chin Ann Math Ser B, 1987, 8: 1–12

    MathSciNet  MATH  Google Scholar 

  38. Kahane J-P. Desintegration des mesures selon la dimension et chaos multiplicatif. In: Harmonic Analysis. Lecture Notes in Mathematics, vol. 1359. Berlin-Heidelberg: Springer, 1988, 199–208

    Chapter  Google Scholar 

  39. Kahane J-P. Random coverings and multiplicative processes. In: Fractal Geometry and Stochastics II. Progress in Probability, vol. 46. Basel: Birkhäuser, 2000, 125–146

    Chapter  Google Scholar 

  40. Kahane J-P, Peyrièere J. Sur certaines martingales de Benoit Mandelbrot. Adv Math, 1976, 22: 131–145

    Article  MathSciNet  MATH  Google Scholar 

  41. Kahane J-P, Salem R. Ensembles parfaits et séries trigonométriques. Paris: Hermann, 1963

    MATH  Google Scholar 

  42. Kolmogorov A N. Précisions sur la structure locale de la turbulence dans un fluide visqueux aux nombres de Reynolds élevés. In: Mécanique de la Turbulence. Paris: CNRS, 1962, 447–451

    Google Scholar 

  43. Liu Q S. On generalized multiplicative cascades. Stochastic Process Appl, 2000, 86: 263–286

    Article  MathSciNet  MATH  Google Scholar 

  44. Liu Q S, Rouault A. Limit theorems for Mandelbrot’s multiplicative cascades. Ann Appl Probab, 2000, 10: 218–239

    Article  MathSciNet  MATH  Google Scholar 

  45. Mandelbrot B B. Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence. In: Statistical Models and Turbulence. Lecture Notes in Physics, vol. 12. Berlin-Heidelberg: Springer, 1972, 333–351

    Chapter  Google Scholar 

  46. Mandelbrot B B. Renewal sets and random cutouts. Z Wahrsch Verw Gebiete, 1972, 22: 145–157

    Article  MathSciNet  MATH  Google Scholar 

  47. Mandelbrot B B. Multiplications aléatoires itérées et distributions invariantes par moyennes pondérée aléatoire. C R Acad Sci Paris Sér A, 1974, 278: 289–292

    MathSciNet  MATH  Google Scholar 

  48. Mandelbrot B B. Multiplications aléatoires itérées et distributions invariantes par moyennes pondérée aléatoire: Quelques extensions. C R Acad Sci Paris Sér A, 1974, 278: 355–358

    MathSciNet  MATH  Google Scholar 

  49. Marcus M B, Pisier G. Random Fourier Series with Applications to Harmonic Analysis. Annals of Mathematics Studies, vol. 101. Princeton: Princeton University Press, 1981

    MATH  Google Scholar 

  50. Paley R E A C, Zygmund A. On some series of functions, (1). Math Proc Cambridge Philos Soc, 1930, 26: 337–357

    Article  MATH  Google Scholar 

  51. Paley R E A C, Zygmund A. On some series of functions, (2). Math Proc Cambridge Philos Soc, 1930, 26: 458–474

    Article  MATH  Google Scholar 

  52. Paley R E A C, Zygmund A. On some series of functions, (3). Math Proc Cambridge Philos Soc, 1932, 28: 190–205

    Article  MATH  Google Scholar 

  53. Rhodes R, Sohier J, Vargas V. Levy multiplicative chaos and star scale invariant random measures. Ann Probab, 2014, 42: 689–724

    Article  MathSciNet  MATH  Google Scholar 

  54. Rhodes R, Vargas V. Gaussian multiplicative chaos and applications: A review. Probab Surv, 2014, 11: 315–392

    Article  MathSciNet  MATH  Google Scholar 

  55. Rhodes R, Vargas V. Lecture notes on Gaussian multiplicative chaos and Liouville quantum gravity. arXiv:1602.07323, 2016

  56. Shamov A. On Gaussian multiplicative chaos. J Funct Anal, 2016, 270: 3224–3261

    Article  MathSciNet  MATH  Google Scholar 

  57. Stein E M, Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton: Princeton University Press, 1971

    MATH  Google Scholar 

  58. Titchmarsh E C. The Theory of the Riemann Zeta-Function, 2nd ed. Oxford: Oxford University Press, 1986

    MATH  Google Scholar 

  59. Waymire E C, Williams S C. A cascade decomposition theory with applications to Markov and exchangeable cascades. Trans Amer Math Soc, 1996, 348: 585–632

    Article  MathSciNet  MATH  Google Scholar 

  60. Wittmann R. A general law of iterated logarithm. Z Wahrsch Verw Gebiete, 1985, 68: 521–543

    Article  MathSciNet  MATH  Google Scholar 

  61. Zygmund A. Trigonometric Series, 3rd ed, Volume I & II Combined. Cambridge: Cambridge University Press, 2002

    MATH  Google Scholar 

Download references

Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant No. 11971192). The authors thank Julien Barral and Vincent Vargas for pointing out the reference [32] in which Janne Junnila obtained the result about the full action on the Lebesgue measure, even for α in a complex domain containing (−2, 2) in the case of Theorem 1.1 and he also studied other fields than the trigonometric field on \(\mathbb{T}\).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Aihua Fan

  2. LAMFA, CNRS UMR 7352, University of Picardie, Amiens, 80039, France

    Aihua Fan

  3. CMLA, ENS-Cachan, CNRS, University Paris-Saclay, Paris, 91190, France

    Yves Meyer

Authors
  1. Aihua Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yves Meyer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aihua Fan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fan, A., Meyer, Y. Trigonometric multiplicative chaos and applications to random distributions. Sci. China Math. 66, 3–36 (2023). https://doi.org/10.1007/s11425-021-1969-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-021-1969-3

Keywords

MSC(2020)

Search

Navigation