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Infinite Dimensional Multifractal Analysis of the Wiener measure

Aihua Fan, Mathieu Helfter

TL;DR

The paper develops a scale-based, infinite-dimensional multifractal framework by proving a general Frostman Lemma for arbitrary Hausdorff gauges on broad spaces and establishing a mass-distribution principle at this level. It then applies the formalism to the Wiener measure, identifying critical Hölder-regular trajectories and deriving a precise local-order spectrum tied to Hölder and Orey exponents, in particular showing $\mathsf{ord}_{\mathrm{loc}}W(\omega)=2(α^{-1}-1)$ for $\omega\in C_α^{\mathrm{crit}}$ and a corresponding spectrum $f_{\mathrm{crit}}(ξ)=1+ξ/2$ for $ξ\ge2$. The results bridge small-ball asymptotics, cylinder estimates, and Gaussian-process regularity to characterize nontypical trajectories in infinite dimensions, offering a robust framework for extending multifractal analysis beyond finite-dimensional settings. These insights pave the way for applying scale-based multifractal methods to broader Gaussian processes and to theoretical questions about path-space geometry and thermodynamical formalisms. $\,$

Abstract

We present a multifractal formalism for measures on infinite dimensional metric spaces, in terms of scales instead of dimensions in the classical multifractal analysis. We prove a multifractal formalism with a suitable scaling, called order, for the Wiener measure, which is the probability law of the standard Brownian motion. We also prove the fundamental Frostman Lemma on a large class of Polish spaces, for which the increasing sets lemma holds.

Infinite Dimensional Multifractal Analysis of the Wiener measure

TL;DR

The paper develops a scale-based, infinite-dimensional multifractal framework by proving a general Frostman Lemma for arbitrary Hausdorff gauges on broad spaces and establishing a mass-distribution principle at this level. It then applies the formalism to the Wiener measure, identifying critical Hölder-regular trajectories and deriving a precise local-order spectrum tied to Hölder and Orey exponents, in particular showing for and a corresponding spectrum for . The results bridge small-ball asymptotics, cylinder estimates, and Gaussian-process regularity to characterize nontypical trajectories in infinite dimensions, offering a robust framework for extending multifractal analysis beyond finite-dimensional settings. These insights pave the way for applying scale-based multifractal methods to broader Gaussian processes and to theoretical questions about path-space geometry and thermodynamical formalisms.

Abstract

We present a multifractal formalism for measures on infinite dimensional metric spaces, in terms of scales instead of dimensions in the classical multifractal analysis. We prove a multifractal formalism with a suitable scaling, called order, for the Wiener measure, which is the probability law of the standard Brownian motion. We also prove the fundamental Frostman Lemma on a large class of Polish spaces, for which the increasing sets lemma holds.

Paper Structure

This paper contains 25 sections, 24 theorems, 212 equations.

Table of Contents

  1. Introduction
  2. A general Frostman Lemma
  3. Scalings and scales
  4. Multifractal formalism
  5. Multifractal formalism of the Wiener measure
  6. A multifractal formalism in infinite dimensional spaces
  7. Scalings
  8. Hausdorff, packing and local scales
  9. Level sets and spectra
  10. Generalized Frostman Lemma and mass distribution principle
  11. Frostman Lemma for general Hausdorff functions on separable metric spaces
  12. Mass distribution principle and its application to level sets
  13. Multifractal analysis of the Wiener measure
  14. Local order of Wiener measure and Orey exponents of fractional Brownian motions
  15. From small ball estimates to cylinders estimates
  16. ...and 10 more sections

Key Result

Theorem A

Let $(X,d)$ be a $\sigma$-compact metric space and $A \subset X$ a Borel subset. Then for every Hausdorff function $h$ such that $\mathcal{H}^h(A) > 0$, there exist a compact set $K\subset A$, a probability Borel measure $\mu$ supported by $K$, and a constant $C\in(0,\infty)$ such that

Theorems & Definitions (52)

  • Definition 1.1: Hausdorff function
  • Theorem A: Frostman Lemma
  • Theorem B: Mass distribution principle
  • Corollary 1.2
  • Proposition 1.3
  • Definition 1.4: Critical Hölder functions
  • Theorem C
  • Theorem D: Local order of Wiener measure
  • Theorem E
  • Definition 2.1: Scaling helfter2025scales
  • ...and 42 more