Thin Points of Brownian Motion Intersection Local Times

Abstract

Let \(\ell \) be the projected intersection local time of two independent Brownian paths in \(\mathbb{R}^d \) for d = 2, 3. We determine the lower tail of the random variable \(\ell \)(B(0, 1)), where B(0, 1) is the unit ball. The answer is given in terms of intersection exponents, which are explicitly known in the case of planar Brownian motion. We use this result to obtain the multifractal spectrum, or spectrum of thin points, for the intersection local times.