The volume measure of the Brownian sphere is a Hausdorff measure

TL;DR

This paper proves that the volume measure of the Brownian sphere aligns with a specific Hausdorff measure, revealing that its metric structure uniquely determines its volume measure.

Contribution

It establishes the equivalence between the Brownian sphere's volume measure and a particular Hausdorff measure, linking metric and measure-theoretic properties.

Findings

Volume measure equals a constant multiple of a specific Hausdorff measure.

Derived precise estimates on moments of ball volumes in the Brownian sphere.

Showed the volume measure is determined by the metric structure.

Abstract

We prove that the volume measure of the Brownian sphere is equal to a constant multiple of the Hausdorff measure associated with the gauge function $h(r)=r^4\log\log(1/r)$. This shows in particular that the volume measure of the Brownian sphere is determined by its metric structure. As a key ingredient of our proofs, we derive precise estimates on moments of the volume of balls in the Brownian sphere.