Local fluctuations of critical Mandelbrot cascades

TL;DR

This paper studies the local fluctuations of the limiting measures in critical Mandelbrot cascades, providing optimal bounds on measure fluctuations at small scales.

Contribution

It offers new insights into the local behavior of critical Mandelbrot cascade measures, including precise fluctuation estimates at small scales.

Findings

Established optimal bounds for measure fluctuations at small scales.

Analyzed the behavior of the limiting measure at critical temperature.

Provided a detailed description of local measure fluctuations.

Abstract

We investigate so-called generalized Mandelbrot cascades at the freezing (critical) temperature. It is known that, after a proper rescaling, a~sequence of multiplicative cascades converges weakly to some continuous random measure. Our main question is how the limiting measure $\mu$ fluctuates. For any given point $x$, denoting by $B_n(x)$ the ball of radius $2^{-n}$ centered around $x$, we present optimal lower and upper estimates of $\mu(B_n(x))$ as $n \to \infty$.