The volume measure of the Brownian sphere is a Hausdorff measure

2021/05/12 by Gall, Jean-François Le
#60D05 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2105.05615

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)=r⁴loglog(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.

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