Brownian intersection local times: Upper tail asymptotics and thick points

摘要

We equip the intersection of p independent Brownian paths in R-d, d greater than or equal to 2, with the natural measure l defined by projecting the intersection local time measure via one of the Brownian motions onto the set of intersection points. Given a bounded domain U subset of R-d we show that, as a up arrow infinity, the probability of the event {l(U) > a} decays with an exponential rate of a(1)/ptheta, where theta is described in terms of a variational problem. In the important special case when U is the unit ball in R-3 and p = 2, we characterize theta in terms of an ordinary differential equation. We apply our results to the problem of finding the Hausdorff dimension spectrum for the thick points of the intersection of two independent Brownian paths in R-3. [References: 25]