The Annals of Probability

Brownian intersection local times: Upper tail asymptotics and thick points

Wolfgang König and Peter Mörters

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Abstract

We equip the intersection of p independent Brownian paths in $\mathbb{R}^d$, $d\ge 2$, with the natural measure $\ell$ defined by projecting the intersection local time measure via one of the Brownian motions onto the set of intersection points. Given a bounded domain $U\subset\mathbb{R}^d$ we show that, as $a\uparrow\infty$, the probability of the event $\{\ell(U)>a\}$ decays with an exponential rate of $a^{1/p}\theta$, where $\theta$ is described in terms of a variational problem. In the important special case when U is the unit ball in $\mathbb{R}^3$ 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 $\mathbb{R}^3$.

Article information

Source
Ann. Probab., Volume 30, Number 4 (2002), 1605-1656.

Dates
First available in Project Euclid: 10 December 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1039548368

Digital Object Identifier
doi:10.1214/aop/1039548368

Mathematical Reviews number (MathSciNet)
MR1944002

Zentralblatt MATH identifier
1032.60073

Subjects
Primary: 60J65: Brownian motion [See also 58J65] 60G17: Sample path properties 60J55: Local time and additive functionals

Keywords
Brownian motion intersection of Brownian paths intersection local time Wiener sausage upper tail asymptotics Hausdorff measure thick points Hausdorff dimension spectrum multifractal spectrum

Citation

König, Wolfgang; Mörters, Peter. Brownian intersection local times: Upper tail asymptotics and thick points. Ann. Probab. 30 (2002), no. 4, 1605--1656. doi:10.1214/aop/1039548368. https://projecteuclid.org/euclid.aop/1039548368


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