Problèmes de recouvrement et points exceptionnels pour la marche aléatoire et le mouvement brownien
Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 951, pp. 469-480.

La marche aléatoire (ou marche au hasard) est un objet fondamental de la théorie des probabilités. Un des problèmes les plus intéressants pour la marche aléatoire (ainsi que pour le mouvement brownien, son analogue dans un contexte continu) est de savoir comment elle recouvre des ensembles où se trouvent les points qui sont souvent (ou au contraire, rarement) visités, et combien il y a de tels points. Les travaux de Dembo, Peres, Rosen et Zeitouni permettent de résoudre plusieurs conjectures importantes liées à ces questions.

Random walk is a fundamental object in probability theory. One of the most interesting problems for random walk (as well as for brownian motion, its continuous-time analogue) is to know how it covers various sets, where the frequently/rarely visited points lie, and whether there are many such points. Dembo, Peres, Rosen and Zeitouni solve several important open problems related to these questions.

Classification : 60G50, 60J65, 60J55, 28A80
Mot clés : problème de recouvrement, point favori, point épais, point fin, point tardif, analyse multi-fractale, mesure d'occupation, arbre, marche aléatoire, mouvement brownien
Keywords: covering problem, favourite point, thick point, thin point, late point, multifractal analysis, occupation measure, tree, random walk, brownian motion
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Shi, Zhan. Problèmes de recouvrement et points exceptionnels pour la marche aléatoire et le mouvement brownien, dans Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 951, pp. 469-480. http://www.numdam.org/item/SB_2004-2005__47__469_0/

[1] D. Aldous - Probability approximations via the Poisson clumping heuristic, Applied Mathematical Sciences, vol. 77, Springer-Verlag, New York, 1989. | DOI | MR | Zbl

[2] Z. Ciesielski & S. J. Taylor - “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), p. 434-450. | DOI | MR | Zbl

[3] O. Daviaud - “ Extremes of the discrete two-dimensional Gaussian free field”, Ann. Probab. 34 (2006), no. 3, p. 962-986. | MR | Zbl

[4] A. Dembo - “Favorite points, cover times and fractals”, in Lectures on probability theory and statistics, Lecture Notes in Math., vol. 1869, Springer, Berlin, 2005, p. 1-101. | MR | Zbl

[5] A. Dembo, Y. Peres & J. Rosen - “How large a disc is covered by a random walk in n steps ?”, prépublication. | DOI | Zbl

[6] -, “Brownian motion on compact manifolds : cover time and late points”, Electron. J. Probab. 8 (2003), no. 15, 14 pp. (electronic). | EuDML | MR | Zbl

[7] A. Dembo, Y. Peres, J. Rosen & O. Zeitouni - “Late points for random walks in two dimensions”, Ann. Probab. 34 (2006), no. 1, p. 219-263. | MR | Zbl

[8] -, “Thick points for transient symmetric stable processes”, Electron. J. Probab. 4 (1999), no. 10, 13 pp. (electronic). | EuDML | MR | Zbl

[9] -, “Thick points for spatial Brownian motion : multifractal analysis of occupation measure”, Ann. Probab. 28 (2000), no. 1, p. 1-35. | MR | Zbl

[10] -, “Thin points for Brownian motion”, Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 6, p. 749-774. | EuDML | Numdam | MR | Zbl

[11] -, “Thick points for planar Brownian motion and the Erdős-Taylor conjecture on random walk”, Acta Math. 186 (2001), no. 2, p. 239-270. | MR | Zbl

[12] -, “Thick points for intersections of planar sample paths”, Trans. Amer. Math. Soc. 354 (2002), no. 12, p. 4969-5003. | MR | Zbl

[13] -, “Cover times for Brownian motion and random walks in two dimensions”, Ann. of Math. (2) 160 (2004), no. 2, p. 433-464. | MR | Zbl

[14] U. Einmahl - “Extensions of results of Komlós, Major, and Tusnády to the multivariate case”, J. Multivariate Anal. 28 (1989), no. 1, p. 20-68. | MR | Zbl

[15] P. Erdős & P. Révész - “On the favourite points of a random walk”, in Mathematical structure- computational mathematics - mathematical modelling, 2, Publ. House Bulgar. Acad. Sci., Sofia, 1984, p. 152-157. | MR | Zbl

[16] -, “Three problems on the random walk in Z d , Studia Sci. Math. Hungar. 26 (1991), p. 309-320. | MR | Zbl

[17] P. Erdős & S. J. Taylor - “Some problems on the structure of random walk paths”, Acta Math. Sci. Hungar. 11 (1960), p. 137-162. | DOI | MR | Zbl

[18] J. Komlós, P. Major & G. Tusnády - “An approximation of partial sums of independent RV's, and the sample DF. II”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 34 (1976), no. 1, p. 33-58. | MR | Zbl

[19] G. F. Lawler - “On the covering time of a disc by simple random walk in two dimensions”, in Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992), Progr. Probab., vol. 33, Birkhäuser Boston, Boston, MA, 1993, p. 189-207. | MR | Zbl

[20] J.-F. Le Gall - “Some properties of planar Brownian motion”, in École d'Été de Probabilités de Saint-Flour XX-1990, Lecture Notes in Math., vol. 1527, Springer, Berlin, 1992, p. 111-235. | MR | Zbl

[21] E. A. Perkins & S. J. Taylor - “Uniform measure results for the image of subsets under Brownian motion”, Probab. Theory Related Fields 76 (1987), no. 3, p. 257-289. | MR | Zbl

[22] D. Ray - “Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion”, Trans. Amer. Math. Soc. 106 (1963), p. 436-444. | DOI | MR | Zbl

[23] P. Révész - “Clusters of a random walk on the plane”, Ann. Probab. 21 (1993), no. 1, p. 318-328. | MR | Zbl

[24] -, “Covering problems”, Theory Probab. Appl. 38 (1993), p. 367-379. | MR | Zbl

[25] -, Random walk in random and non-random environments, second 'ed., World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. | MR | Zbl

[26] J. Rosen - “A random walk proof of the Erdős-Taylor conjecture”, Period. Math. Hungar. 50 (2005), no. 1-2, p. 223-245. | MR | Zbl

[27] S. J. Taylor - “Regularity of irregularities on a Brownian path”, Ann. Inst. Fourier 24 (1974), p. 195-203. | DOI | EuDML | Numdam | MR | Zbl

[28] B. Tóth - “No more than three favorite sites for simple random walk”, Ann. Probab. 29 (2001), no. 1, p. 484-503. | MR | Zbl

[29] W. Werner - “Random planar curves and Schramm-Loewner evolutions”, in Lectures on probability theory and statistics, Lecture Notes in Math., vol. 1840, Springer, Berlin, 2004, p. 107-195. | MR | Zbl