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Thick points for intersections of planar sample paths. (English) Zbl 1007.60077
Summary: Let \(L_n^{X}(x)\) denote the number of visits to \(x \in \mathbf{Z} ^2\) of the simple planar random walk \(X\), up to step \(n\). Let \(X'\) be another simple planar random walk independent of \(X\). We show that for any \(0<b<1/(2 \pi)\), there are \(n^{1-2\pi b+o(1)}\) points \(x \in \mathbf{Z}^2\) for which \(L_n^{X}(x)L_n^{X'}(x)\geq b^2 (\log n)^4\). This is the discrete counterpart of our main result, that for any \(a<1\), the Hausdorff dimension of the set of thick intersection points \(x\) for which \(\limsup_{r \rightarrow 0} \mathcal{I} (x,r)/(r^2|\log r|^4)=a^2\), is almost surely \(2-2a\). Here \(\mathcal{I}(x,r)\) is the projected intersection local time measure of the disc of radius \(r\) centered at \(x\) for two independent planar Brownian motions run until time \(1\). The proofs rely on a “multi-scale refinement” of the second moment method. In addition, we also consider analogous problems where we replace one of the Brownian motions by a transient stable process, or replace the disc of radius \(r\) centered at \(x\) by \(x+rK\) for general sets \(K\).

MSC:
60J55 Local time and additive functionals
60J65 Brownian motion
28A80 Fractals
60G50 Sums of independent random variables; random walks
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[1] Richard F. Bass and Davar Khoshnevisan, Intersection local times and Tanaka formulas, Ann. Inst. H. Poincaré Probab. Statist. 29 (1993), no. 3, 419 – 451 (English, with English and French summaries). · Zbl 0798.60072
[2] Amir Dembo, Yuval Peres, Jay Rosen, and Ofer Zeitouni, Thick points for spatial Brownian motion: multifractal analysis of occupation measure, Ann. Probab. 28 (2000), no. 1, 1 – 35. · Zbl 1130.60311 · doi:10.1214/aop/1019160110 · doi.org
[3] A. Dembo, Y. Peres, J. Rosen and O. Zeitouni, Thin points for Brownian Motion, Annales de L’Institut Henri Poincaré, 36 (2000), 749-774. CMP 2001:05 · Zbl 0977.60073
[4] Amir Dembo, Yuval Peres, Jay Rosen, and Ofer Zeitouni, Thick points for transient symmetric stable processes, Electron. J. Probab. 4 (1999), no. 10, 13. · Zbl 0927.60077 · doi:10.1214/EJP.v4-47 · doi.org
[5] A. Dembo, Y. Peres, J. Rosen and O. Zeitouni, Thick points for planar Brownian motion and the Erdos-Taylor conjecture on random walk, Acta Math. 186 (2001), 239-270. · Zbl 1008.60063
[6] Uwe Einmahl, Extensions of results of Komlós, Major, and Tusnády to the multivariate case, J. Multivariate Anal. 28 (1989), no. 1, 20 – 68. · Zbl 0676.60038 · doi:10.1016/0047-259X(89)90097-3 · doi.org
[7] P. J. Fitzsimmons and Jim Pitman, Kac’s moment formula and the Feynman-Kac formula for additive functionals of a Markov process, Stochastic Process. Appl. 79 (1999), no. 1, 117 – 134. · Zbl 0962.60067 · doi:10.1016/S0304-4149(98)00081-7 · doi.org
[8] Kiyosi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Springer-Verlag, Berlin-New York, 1974. Second printing, corrected; Die Grundlehren der mathematischen Wissenschaften, Band 125. · Zbl 0837.60001
[9] Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. · Zbl 0571.60002
[10] Robert Kaufman, Une propriété métrique du mouvement brownien, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A727 – A728 (French). · Zbl 0174.21401
[11] J.-F. Le Gall, The exact Hausdorff measure of Brownian multiple points, Seminar on stochastic processes, 1986 (Charlottesville, Va., 1986) Progr. Probab. Statist., vol. 13, Birkhäuser Boston, Boston, MA, 1987, pp. 107 – 137.
[12] W. König and P. Mörters, Brownian intersection local times: upper tail asymptotics and thick points, Preprint (2001). To appear, Ann. Probab. (2002). · Zbl 1032.60073
[13] Gregory F. Lawler, Intersections of random walks, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1991. · Zbl 1228.60004
[14] Jean-François Le Gall, Some properties of planar Brownian motion, École d’Été de Probabilités de Saint-Flour XX — 1990, Lecture Notes in Math., vol. 1527, Springer, Berlin, 1992, pp. 111 – 235. · Zbl 0779.60068 · doi:10.1007/BFb0084700 · doi.org
[15] J.-F. Le Gall, The exact Hausdorff measure of Brownian multiple points, Seminar on stochastic processes, 1986 (Charlottesville, Va., 1986) Progr. Probab. Statist., vol. 13, Birkhäuser Boston, Boston, MA, 1987, pp. 107 – 137. Jean-François Le Gall, The exact Hausdorff measure of Brownian multiple points. II, Seminar on Stochastic Processes, 1988 (Gainesville, FL, 1988) Progr. Probab., vol. 17, Birkhäuser Boston, Boston, MA, 1989, pp. 193 – 197.
[16] Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. · Zbl 0819.28004
[17] Edwin A. Perkins and S. James Taylor, Uniform measure results for the image of subsets under Brownian motion, Probab. Theory Related Fields 76 (1987), no. 3, 257 – 289. · Zbl 0613.60071 · doi:10.1007/BF01297485 · doi.org
[18] Pál Révész, Random walk in random and nonrandom environments, World Scientific Publishing Co., Inc., Teaneck, NJ, 1990. · Zbl 0733.60091
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