The average density of the path of planar Brownian motion

We show that the occupation measure μ on the path of a planar Brownian motion run for an arbitrary finite time interval has an average density of order three with respect to the gauge function ϕ(t)=t2log(1/t). In other words, almost surely,limε↓01log|log ε|∫ε1/eμ(B(x,t))ϕ(t)d t|t log t|=2 at μ-almost every x.We also prove a refinement of this statement: Almost surely, at μ-almost every x,limε↓01log|logε|∫ε1/eδμ(B(x,t))ϕ(t)d t|t log t|=∫0∞δ{a}ae−ada,in other words, the distribution of the ϕ-density function under the averaging measures of order three converges to a gamma distribution with parameter two.