# zbMATH — the first resource for mathematics

Thick points for spatial Brownian motion: multifractal analysis of occupation measure. (English) Zbl 1130.60311
Summary: Let $$\mathcal T(x,r)$$ denote the total occupation measure of the ball of radius $$r$$ centered at $$x$$ for Brownian motion in $$\mathbb{R}^3$$. We prove that $$\sup_{|x|\leq1}\mathcal T(x,r)/(r^{2}|\log r|)\rightarrow16/\pi^2$$ a.s. as $$r\rightarrow0$$, thus solving a problem posed by Taylor in 1974. Furthermore, for any $$a \in(0,16/\pi^2)$$, the Hausdorff dimension of the set of “thick points” $$x$$ for which $$\lim\sup_{r \to 0}\mathcal T(x,r)/(r^2|\log r|) = a$$ is almost surely $$2-a\pi^2/8$$; this is the correct scaling to obtain a nondegenerate “multifractal spectrum” for Brownian occupation measure. Analogous results hold for Brownian motion in any dimension $$d \geq 3$$. These results are related to the LIL of Ciesielski and Taylor for the Brownian occupation measure of small balls in the same way that Lévy’s uniform modulus of continuity, and the formula of Orey and Taylor for the dimension of “fast points” are related to the usual LIL. We also show that the lim inf scaling of $$\mathcal T(x,r)$$ is quite different: we exhibit nonrandom $$c_1,c_2 \geq 0$$, such that $$c_1 < \sup_x\lim \inf _{r \to 0}\mathcal T(x,y)/r^2 < c_2$$ a.s. In the course of our work we provide a general framework for obtaining lower bounds on the Hausdorff dimension of random fractals of “limsup type”.

##### MSC:
 60J65 Brownian motion 28A78 Hausdorff and packing measures 28A80 Fractals 60F15 Strong limit theorems 60J55 Local time and additive functionals
Full Text:
##### References:
 [1] Barlow, M. T. and Perkins, E. (1984). Levels at which every Brownian excursion is exceptional. Seminar on Probability XVIII. Lecture Notes in Math. 1059 1-28. Springer, Berlin. · Zbl 0555.60050 · numdam:SPS_1984__18__1_0 · eudml:113482 [2] Benjamini, I. and Peres, Y. (1994). Tree-indexed random walks on groups and first passage percolation. Probab. Theory Related Fields 98 91-112. · Zbl 0794.60068 · doi:10.1007/BF01311350 [3] Ciesielski,and Taylor, S. J. (1962). First passage and sojourn times and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc. 103 434-452. JSTOR: · Zbl 0121.13003 · doi:10.2307/1993838 · links.jstor.org [4] Deheuvels, P. and Mason, D. M. (1998). Random fractal functional laws of the iterated logarithm. Studia Sci. Math Hungar. 34 89-106. · Zbl 0916.60037 [5] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (1999). Thick points for transient symmetric stable processes. Elect. J. Probab. 4 1-18. · Zbl 0927.60077 · emis:journals/EJP-ECP/EjpVol4/paper9.abs.html · eudml:120048 [6] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (1999) Thick points for planar Brownian motion and the Erd os-Taylor conjecture on random walk. Acta Math. · Zbl 1008.60063 [7] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (1999). Thin points for Brownian motion. Ann. Inst. H. Poincaré Math. Statist. Probab. · Zbl 0977.60073 [8] Falconer, K. J. (1990). Fractal Geometry: Mathematical Foundations and Applications. Wiley, NewYork. · Zbl 0689.28003 [9] Hu, X. and Taylor, S. J. (1997). The multifractal structure of stable occupation measure. Stochastic Process. Appl. 66 283-299. · Zbl 0888.28004 · doi:10.1016/S0304-4149(97)00127-0 [10] Kaufman, R. (1969). Une propriété metriqué du mouvement brownien. C. R. Acad. Sci. Paris 268 727-728. · Zbl 0174.21401 [11] Lawler, G. The frontier of a Brownian path is multifractal. [12] Olsen, L. (1995). A multifractal formalism. Adv. Math. 116 82-196. · Zbl 0841.28012 · doi:10.1006/aima.1995.1066 [13] Orey, S. and Taylor, S. J. (1974). Howoften on a Brownian path does the lawof the iterated logarithm fail? Proc. London Math. Soc. 28 174-192. · Zbl 0292.60128 · doi:10.1112/plms/s3-28.1.174 [14] Pemantle, R. and Peres, Y. (1995). Galton-Watson trees with the same mean have the same polar sets. Ann. Probab. 23 1102-1124. · Zbl 0833.60085 · doi:10.1214/aop/1176988175 [15] Pemantle, R., Peres, Y. and Shapiro, J. W. (1996). The trace of spatial Brownian motion is capacity-equivalent to the unit square. Probab. Theory Related Fields 106 379-399. · Zbl 0864.60065 · doi:10.1007/s004400050070 [16] Peres, Y. (1996). Intersection-equivalence of Brownian paths and certain branching processes. Comm. Math. Phys. 177 417-443. · Zbl 0851.60080 · doi:10.1007/BF02101900 [17] Perkins, E. A. (1983). On the Hausdorff dimension of Brownian slow points.Wahrsch. Verw. Gebeite 64 369-399. · Zbl 0506.60079 · doi:10.1007/BF00532968 [18] Perkins, E. A. and Taylor, S. J. (1987). Uniform measure results for the image of subsets under Brownian motion. Probab. Theory Related Fields 76 257-289. · Zbl 0613.60071 · doi:10.1007/BF01297485 [19] Perkins, E. A. and Taylor, S. J. (1988). Measuring close approaches on a Brownian path. Ann. Probab. 16 1458-1480. · Zbl 0659.60113 · doi:10.1214/aop/1176991578 [20] Perkins, E. A. and Taylor, S. J. (1998). The multifractal structure of super-Brownian motion. Ann. Inst. H. Poincaré Math. Statist. Probab. 34 97-138. · Zbl 0905.60031 · doi:10.1016/S0246-0203(98)80020-4 · numdam:AIHPB_1998__34_1_97_0 · eudml:77597 [21] Revuz D. and Yor, M. (1991) Continuous Martingales and Brownian Motion. Springer, New York. · Zbl 0731.60002 [22] Reidi, R. (1995). An improved multifractal formalism and self-similar measures. J. Math. Anal. Appl. 189 462-490. · Zbl 0819.28008 · doi:10.1006/jmaa.1995.1030 [23] Rogers, C. A. and Taylor, S. J. (1961). Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8 1-31. · Zbl 0145.28701 · doi:10.1112/S0025579300002084 [24] Shieh, N.-R. and Taylor, S. J. (1998). Logarithmic multifractal spectrum of stable occupation measure. Stochastic Process. Appl. 75 249-269. [Correction (1999). Trends in Probability and Related Analysis 147-158. World Scientific, Singapore. · Zbl 0932.60041 [25] Stroock, D. W. (1993). Probability Theory, an Analytic View. Cambridge Univ. Press. · Zbl 0925.60004 [26] Taylor, S. J. (1974). Regularity of irregularities on a Brownian path. Ann. Inst. Fourier (Grenoble) 39 195-203. · Zbl 0262.60059 · doi:10.5802/aif.513 · numdam:AIF_1974__24_2_195_0 · eudml:74172 [27] Taylor, S. J. (1986). The measure theory of random fractals. Math. Proc. Cambridge Philos. Soc. 100 383-406. · Zbl 0622.60021 · doi:10.1017/S0305004100066160 [28] Taylor, S. J. (1995). Super Brownian motion is a fractal measure for which the multifractal formalism is invalid. In Symposium in Honor of Benoit Mandelbrot 3 737-746 (Curagao). · Zbl 0873.60055 · doi:10.1142/S0218348X95000655 [29] Taylor, S. J. and Tricot, C. (1985). Packing measure and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288 679-699. · Zbl 0537.28003 · doi:10.2307/1999958 [30] Tricot, C. (1982). Two definitions of fractal dimension. Math. Proc. Cambridge Philos. Soc. 91 57-74. · Zbl 0483.28010 · doi:10.1017/S0305004100059119 [31] Watson, G. N. (1922). A Treatise on the Theory of Bessel Functions. Cambridge Univ. Press. · JFM 48.0412.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.