Uniform convergence for complex [0, 1]-martingales

Abstract

International audiencePositive T-martingales were developed as a general framework that extends the positive measure-valued martingales and are meant to model intermittent turbulence. We extend their scope by allowing the martingale to take complex values. We focus on martingales constructed on the interval T = [0, 1] and replace random measures by random functions. We specify a large class of such martingales for which we provide a general sufficient condition for almost sure uniform convergence to a nontrivial limit. Such a limit yields new examples of naturally generated multifractal processes that may be of use in multifractal signals modeling

Complex Gaussian Multiplicative Chaos

“…More specifically, if A is a fixed set, the following limit lim ε→0 1 ε ln |M γ,β ε (A)| should be independent from ρ. However, we are not aware of a solid argument which would say that the proper renormalization of M γ,β ε (which is the object of this work in the case ρ = 0) is independent from ρ (except in the inner phase I where it is a consequence of the results of [4,5] in dimension 1). Let us stress that the general case is of course interesting (and natural to look at in the case ρ = 1) but its study requires new ideas beyond the methodology of this paper.…”

Exact dimensionality and projection properties of Gaussian multiplicative chaos measures

“…Assume that D satisfies (A0). If ν is exact-dimensional with dimension α > γ 2 2 , then, the GMC measure ν of ν is well-defined and non-trivial, and almost surely, ν is exact-dimensional with dimension α − γ 2 2 . This corollary applies to the large class of measures that are exact dimensional, including self-similar measures and, more generally, Gibbs measures on self-conformal sets, see [12,14], as well as planar self-affine measures [1].…”

“…and applying the Borel-Cantelli lemma we get that, almost surely, ν(B(x, 2 −n )) ≤ ν(B n (x)) ≤ 9 · 2 −nκ for all sufficiently large n for ν-almost all x (since lim n→∞ ν(∪ S∈S • n S) = ν(D)), where B n (x) = S ′ ∈N (Sn(x)) S ′ , and S n (x) is the square in S n containing x. Thus, almost surely, for ν-almost all x, lim inf n→∞ 1 log 2 −n log ν(B(x, 2 −n )) ≥ κ, for all κ < α − γ 2 2 p, where we may take p arbitrarily close to 1. 3.2.…”

“…The γ-LQG measure µ, obtained from circle averages of GFF acting on planar Lebesgue measure µ on D, is almost surely exact dimensional of dimension dim H µ = 2 − γ 2 2 , in which case for almost all θ the orthogonal projections π θ µ are exact dimensional of dimension min{1, dim H µ} and are absolutely continuous if 0 < γ < √ 2 ≈ 1.4142. Here we show that, for suitable domains D, if 0 < γ < 1 33 858 − 132 √ 34 ≈ 0.28477489 then for all θ simultaneously, not only are the projected measures π θ µ absolutely continuous with respect to Lebesgue measure but also the Radon-Nikodym derivatives are β-Hölder for some β > 0.…”

Mandelbrot's cascades: a legendary destiny

“…The following results, illustrated by Figs. 5.3 and 5.4, are proved in [73,74]. It is natural to study separately the conservative case (for which W 0 + · · · + W b−1 is 1 a.s.) and the non conservative case.…”