The average density of the path of planar Brownian motion

Abstract

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)= t 2log(1/ t). In other words, almost surely, lim ε↓0 1 log| log ε| ∫ ε 1/ e μ(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, lim ε↓0 1 log| logε| ∫ ε 1/ e δ μ(B(x,t)) ϕ(t) d t |t log t| =∫ 0 ∞δ {a}a e −a da, 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.