Infinite Dimensional Multifractal Analysis of the Wiener measure

TL;DR

This paper develops a new multifractal analysis framework for measures in infinite dimensional spaces, applying it to Wiener measure, and establishes foundational lemmas for Polish spaces.

Contribution

It introduces a multifractal formalism based on scales for infinite dimensional measures and proves a Frostman Lemma applicable to a broad class of Polish spaces.

Findings

Established a multifractal formalism for Wiener measure

Proved a Frostman Lemma for Polish spaces

Extended multifractal analysis to infinite dimensional settings

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.