The average density of the path of planar Brownian motion

Stochastic Processes and their Applications

Volumn 74, Issue 1, 1998, Pages 133-149

  Mörters, Peter ^a  

^a Rheinland Pfälzische Technische Universität Kaiserslautern Landau   (Germany)

Author keywords

28A75; 28A80; 60G17; 60J65; Average density; Brownian motion; Density distribution; Logarithmic averages; Occupation measure; Pathwise Kallianpur Robbins law

Indexed keywords

EID: 0038378464     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-4149(97)00118-X     Document Type: Article

References (25)

1. Bandt, C., 1992. The tangent distribution for self-similar measures. Lecture at the 5th Conf. on Real Analysis and Measure Theory, Capri.
2. Bedford T., Fisher A.M. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc. 64(3):1992;95-124.
3. Brosamler G.A. The asymptotic behaviour of certain additive functionals of Brownian motion. Invent. Math, 20:1973;87-96.
4. Ciesielski Z., Taylor S.J. First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc, 103:1962;434-450.
5. Falconer K.J. Wavelet transforms and order-two densities of fractals. J. Statist. Phys. 67:1992;781-793.
6. Falconer K.J., Springer O.B. Order-two density of sets and measures with non-integral dimension. Mathematika. 42:1995;1-14.
7. Falconer K.J., Xiao Y.M. Average densities of the image and zero set of stable processes. Stoch. Proc. Appl. 55:1995;271-283.
8. Fisher A.M. Convex-invariant means and a pathwise central limit theorem. Adv. Math. 63:1987;213-246.
9. Földes, A., 1992. Strong theorems for random walk and its local time. In: Proc. 4th Int. Meeting of Statistics in the Basque Country, pp. 263-267.
10. Graf S. On Bandt's tangential distribution for self-similar measures. Mh. Math. 120:1995;223-246.
11. Kallenberg, O, 1983. Random Measures. Akademie, Berlin.
12. Knight F.B. On the random walk and Brownian motion. Trans. Amer. Math. Soc. 103:1962;218-228.
13. Marcus M.B., Rosen J. Logarithmic averages of the local times of recurrent random walks and Levy processes. Stoch. Proc. Appl. 59:1995;175-184.
14. Marstrand J.M. Order-two density and the strong law of large numbers. Mathematika. 43:1996;1-22.
15. Mattila P. The Geometry of Sets and Measures in Euclidean Spaces. 1995;Cambridge University Press, Cambridge.
16. Mörters, P., Symmetry properties of average densities and tangent measure distributions of measures on the line. Adv. Appl.Math., to appear.
17. Mörters P. Average densities and linear rectifiability of measures. Mathematika. 44:1997;313-324.
18. Mörters, P., Preiss, D., Tangent measure distributions of fractal measures. Preprint, Universität Kaiserslautern. (1997).
19. Patzschke N., Zähle M. Fractional differentiation in the self-affine case I. Random functions. Stoch. Proc. Appl. 43:1992;165-175.
20. Patzschke N., Zähle M. Fractional differentiation in the self-affine case II. Extremal processes. Stoch. Proc. Appl. 45:1993;61-72.
21. Patzschke N., Zähle M. Fractional differentiation in the self-affine case IV. Random measures. Stoch. Stoch. Rep. 49:1994;87-98.
22. 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.
23. Stroock D.W. Probability Theory, an Analytic View. 1993;Cambridge University Press, Cambridge.
24. Taylor S.J. The exact Hausdorff measure on the sample path for planar Brownian motion. Proc. Camb. Phil. Soc. 60:1964;253-258.
25. Zähle U. Self-similar random measures I. Notion, carrying Hausdorff dimension and hyperbolic distribution. Probab. Theory Related Fields. 80:1998;79-100.