Brownian Intersection Local Times : Upper Tail Asymptotics and Thick Points

Konig, Wolfgang; Morters, Peter

HPL-BRIMS-2001-03

Keyword(s): Brownian motion; intersection of Brownian paths; intersection local time; Wiener sausage; upper tail asymptotics; Hausdorff measure; thick points; Hausdorff dimension spectrum; multifractal spectrum

Abstract:Please Note. This abstract contains mathematical formulae which cannot be represented here. We equip the intersection of p independent Brownian paths in R^d, d ≥ 2, with the natural measure e 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 ⊂ R^d we show that, as a ↑ ∞ , the probability of the event {e (U) > a } decays with an exponential rate of a ^1/p θ , where θ is described in terms of a variational problem. In the important special case when U is the unit ball in R³ and p =2, we characterize θ 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 R³. Notes: Peter Morters, Fachbereich Mathematik, Universitaet Kaiserslautern, Erwin- Schroedinger-Str., 67663 Kaiserslautern, Germany

41 Pages

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