2000 character limit reached
Fourier dimension of Mandelbrot multiplicative cascades (2409.13455v3)
Published 20 Sep 2024 in math.PR, math-ph, math.CA, and math.MP
Abstract: We investigate the Fourier dimension, $\dim_F\mu$, of Mandelbrot multiplicative cascade measures $\mu$ on the $d$-dimensional unit cube. We show that if $\mu$ is the cascade measure generated by a sub-exponential random variable then [\dim_F\mu=\min{2,\dim_2\mu}\,,] where $\dim_2\mu$ is the correlation dimension of $\mu$ and it has an explicit formula. For cascades on the circle $S\subset\mathbb{R}2$, we obtain [\dim_F\mu\ge\frac{\dim_2\mu}{2+\dim_2\mu}\,.]
- E. Aïdékon. Convergence in law of the minimum of a branching random walk. Ann. Probab., 41(3A):1362–1426, 2013.
- J. Barral. Continuity of the multifractal spectrum of a random statistically self-similar measure. J. Theoret. Probab., 13(4):1027–1060, 2000.
- Gaussian multiplicative chaos and KPZ duality. Comm. Math. Phys., 323(2):451–485, 2013.
- Critical Mandelbrot cascades. Comm. Math. Phys., 325(2):685–711, 2014.
- I. Benjamini and O. Schramm. KPZ in one dimensional random geometry of multiplicative cascades. Comm. Math. Phys., 289(2):653–662, 2009.
- Harmonic Analysis of Mandelbrot Cascades in the context of vector-valued measures. Preprint, available at arXiv:2409.13164.
- K. Falconer and X. Jin. Exact dimensionality and projection properties of Gaussian multiplicative chaos measures. Trans. Amer. Math. Soc., 372(4):2921–2957, 2019.
- K. J. Falconer and S. Troscheit. Box-counting dimension in one-dimensional random geometry of multiplicative cascades. Comm. Math. Phys., 399(1):57–83, 2023.
- C. Garban and V. Vargas. Harmonic analysis of Gaussian multiplicative chaos on the circle. Preprint, available at arXiv:2311.04027.
- L. Grafakos. Classical Fourier analysis, volume 249 of Graduate Texts in Mathematics. Springer, New York, second edition, 2008.
- Y. Heurteaux. An introduction to Mandelbrot cascades. In New trends in applied harmonic analysis, Appl. Numer. Harmon. Anal., pages 67–105. Birkhäuser/Springer, Cham, 2016.
- How projections affect the dimension spectrum of fractal measures. Nonlinearity, 10(5):1031–1046, 1997.
- B. Jaffuel. The critical barrier for the survival of branching random walk with absorption. Ann. Inst. Henri Poincaré Probab. Stat., 48(4):989–1009, 2012.
- J.-P. Kahane. Fractals and random measures. Bull. Sci. Math., 117(1):153–159, 1993.
- J.-P. Kahane and J. Peyrière. Sur certaines martingales de Benoit Mandelbrot. Advances in Math., 22(2):131–145, 1976.
- R. Lyons. Seventy years of Rajchman measures. In Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993), pages 363–377, 1995.
- T. Madaule. Convergence in law for the branching random walk seen from its tip. J. Theoret. Probab., 30(1):27–63, 2017.
- B. Mandelbrot. Intermittent turbulence in self similar cascades: divergence of high moments and dimension of carrier. J. Fluid Mech., 62:331–333, 1974.
- B. Mandelbrot. Intermittent turbulence and fractal dimension: kurtosis and the spectral exponent 5/3+B53𝐵5/3+B5 / 3 + italic_B. In Turbulence and Navier-Stokes equations (Proc. Conf., Univ. Paris-Sud, Orsay, 1975), volume Vol. 565 of Lecture Notes in Math., pages 121–145. Springer, Berlin-New York, 1976.
- P. Mattila. Fourier analysis and Hausdorff dimension, volume 150 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2015.
- D. Ryou. Near-optimal restriction estimates for Cantor sets on the parabola. Int. Math. Res. Not. IMRN, (6):5050–5099, 2024.
- T. Sahlsten. Fourier transforms and iterated function systems. Preprint, available at arXiv:2311.00585.
- P. Shmerkin and V. Suomala. Spatially independent martingales, intersections, and applications. Mem. Amer. Math. Soc., 251(1195):v+102, 2018.
- R. Vershynin. High-dimensional probability, volume 47 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2018. An introduction with applications in data science, With a foreword by Sara van de Geer.