The exact packing measure of Brownian double points

Abstract

Let \(D\subset {\mathbb{R}}^3\) be the set of double points of a three-dimensional Brownian motion. We show that, if ξ = ξ3(2,2) is the intersection exponent of two packets of two independent Brownian motions, then almost surely, the ϕ-packing measure of D is zero if $$ \int_{0^+} r^{-1-\xi} \phi(r)^{\xi} \, dr < \infty,$$and infinity otherwise. As an important step in the proof we show up-to-constants estimates for the tail at zero of Brownian intersection local times in dimensions two and three.