Electronic Journal of Probability

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

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1457125518

Digital Object Identifier
doi:10.1214/EJP.v4-46

Mathematical Reviews number (MathSciNet)
MR1690313

Zentralblatt MATH identifier
0937.60032

Rights

Citation

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

References

• Bedford, T., and Fisher, A.M., Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc.(3). 64 (1992) 95–124.
• Dynkin, E.B., Additive functionals of several time–reversible Markov processes. J. Funct. Anal. 42 (1981) 64–101.
• Falconer, K.J., Techniques in fractal geometry. Wiley, Chichester, 1997.
• Falconer, K.J., Wavelet transforms and order–two densities of fractals. Journ. Stat. Phys., 67 (1992) 781–793.
• Falconer, K.J., and Springer, O.B., Order–two density of sets and measures with non–integral dimension. Mathematika. 42 (1995) 1–14.
• Falconer, K.J., and Xiao, Y., Average densities of the image and zero set of stable processes. Stoch. Proc. Appl. 55 (1995) 271–283.
• Geman, D., Horowitz, J., and Rosen, J., A local time analysis of intersections of Brownian motion in the plane. Ann. Probab. 12 (1984) 86–107.
• Kallenberg, O., Random Measures. Akademie-Verlag, Berlin, 1983.
• Le Gall, J.F., Sur la saucisse de Wiener et les points multiples du mouvement brownien. Ann. Probab. 14 (1986) 1219–1244.
• Le Gall, J.F., The exact Hausdorff measure of Brownian multiple points I and II. In: Seminar on Stochastic Processes 1986, 107–137, Birkhäuser, Boston 1987 and Seminar on Stochastic Processes 1988, 193–197, Birkhäuser, Boston 1989.
• Le Gall, J.F., Some properties of planar Brownian motion. In: Lecture Notes in Math. Vol. 1527, Springer Verlag (New York) 1992.
• Le Gall, J.F., and Taylor, S.J., The packing measure of planar Brownian motion. In: Seminar on Stochastic Processes 1986, 139–147, Birkhäuser, Boston 1987.
• Leistritz, L., Ph.D. Dissertation, University of Jena (1994).
• Marstrand, J.M., Order–two density and the strong law of large numbers. Mathematika. 43 (1996) 1–22.
• Mandelbrot, B.B., Measures of fractal lacunarity: Minkowski content and alternatives. In: Bandt, Graf, Zähle (Eds.), Fractal Geometry and Stochastics, 15–42, Birkhäuser (Basel) 1995.
• Mattila, P., The Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge, 1995.
• Mecke, J., Stationäre zufällige Maße auf lokalkompakten abelschen Gruppen. Zeitschr. f. Wahrsch. verw. Gebiete, 9 (1967) 36–58.
• Mörters, P., Average densities and linear rectifiability of measures. Mathematika 44 (1997) 313-324.
• Mörters, P., Symmetry properties of average densities and tangent measure distributions of measures on the line. Adv. Appl. Math. 21 (1998) 146–179.
• Mörters, P., The average density of the path of planar Brownian motion. Stoch. Proc. Appl. 74 (1998) 133–149.
• Mörters, P., and Preiss, D., Tangent measure distributions of fractal measures. Math. Ann. 312 (1998) 53–93.
• Patzschke, N., and Zähle, M., Fractional differentiation in the self–affine case IV. Random measures. Stoch. Stoch. Rep. 49 (1994) 87–98.
• Ray, D., Sojourn times and the exact Hausdorff measure of the sample paths of planar Brownian motion. Trans. Amer. Math. Soc. 108 (1963) 436–444.
• Shieh, N.R., A growth condition for Brownian intersection points. In: Trends in Probability and Related Analysis, Proceedings of SAP'96, 265–272, World Scientific Singapore 1997.
• Taylor, S.J., The measure theory of random fractals. Math. Proc. Camb. Phil. Soc. 100 (1986) 383–486.
• Zähle, U., Self-similar random measures I. Notion, carrying Hausdorff dimension and hyperbolic distribution. Prob. Th. Rel. Fields. 80 (1988) 79–100.