Density theorems for the intersection local times of planar Brownian motion

• We show that the intersection local times $$\mu_p$$ on the intersection of $$p$$ independent planar Brownian paths have an average density of order three with respect to the gauge function $$r^2\pi\cdot (log(1/r)/\pi)^p$$, more precisely, almost surely, $\lim\limits_{\varepsilon\downarrow 0} \frac{1}{log |log\ \varepsilon|} \int_\varepsilon^{1/e} \frac{\mu_p(B(x,r))}{r^2\pi\cdot (log(1/r)/\pi)^p} \frac{dr}{r\ log (1/r)} = 2^p \mbox{ at \mu_p-almost every x.}$ We also show that the lacunarity distributions of $$\mu_p$$, at $$\mu_p$$-almost every point, is given as the distribution of the product of $$p$$ independent gamma(2)-distributed random variables. The main tools of the proof are a Palm distribution associated with the intersection local time and an approximation theorem of Le Gall.

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