Electronic Journal of Probability

Small Scale Limit Theorems for the Intersection Local Times of Brownian Motion

Peter Mörters and Narn-Rueih Shieh

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In this paper we contribute to the investigation of the fractal nature of the intersection local time measure on the intersection of independent Brownian paths. We particularly point out the difference in the small scale behaviour of the intersection local times in three-dimensional space and in the plane by studying almost sure limit theorems motivated by the notion of average densities introduced by Bedford and Fisher. We show that in 3-space the intersection local time measure of two paths has an average density of order two with respect to the gauge function $\varphi(r)=r$, but in the plane, for the intersection local time measure of p Brownian paths, the average density of order two fails to converge. The average density of order three, however, exists for the gauge function $\varphi_p(r)=r^2[\log(1/r)]^p$. We also prove refined versions of the above results, which describe more precisely the fluctuations of the volume of small balls around these gauge functions by identifying the density distributions, or lacunarity distributions, of the intersection local times.

Article information

Electron. J. Probab., Volume 4 (1999), paper no. 9, 23 pp.

Accepted: 23 April 1999
First available in Project Euclid: 4 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G17: Sample path properties
Secondary: 60J65: Brownian motion [See also 58J65] 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] 28A80: Fractals [See also 37Fxx]

Brownian motion intersection local time Palm distribution average density densitydistribution lacunarity distribution logarithmicaverage

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Mörters, Peter; Shieh, Narn-Rueih. Small Scale Limit Theorems for the Intersection Local Times of Brownian Motion. Electron. J. Probab. 4 (1999), paper no. 9, 23 pp. doi:10.1214/EJP.v4-46. https://projecteuclid.org/euclid.ejp/1457125518

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