In this paper we contribute to the investigation of the fractal nature of the intersection local time measure on the intersection of independent Brownian paths. We particularly point out the difference in the small scale behaviour of the intersection local times in three-dimensional space and in the plane by studying almost sure limit theorems motivated by the notion of average densities introduced by Bedford and Fisher. We show that in 3-space the intersection local time measure of two paths has an average density of order two with respect to the gauge function $\varphi(r)=r$, but in the plane, for the intersection local time measure of p Brownian paths, the average density of order two fails to converge. The average density of order three, however, exists for the gauge function $\varphi_p(r)=r^2[\log(1/r)]^p$. We also prove refined versions of the above results, which describe more precisely the fluctuations of the volume of small balls around these gauge functions by identifying the density distributions, or lacunarity distributions, of the intersection local times.
"Small Scale Limit Theorems for the Intersection Local Times of Brownian Motion." Electron. J. Probab. 4 1 - 23, 1999. https://doi.org/10.1214/EJP.v4-46