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
Mots clés : problème de recouvrement, point favori, point épais, point fin, point tardif, analyse multi-fractale, mesure d'occupation, arbre, marche aléatoire, mouvement brownien
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     author = {Shi, Zhan},
     title = {Probl\`emes de recouvrement et points exceptionnels pour la marche al\'eatoire et le mouvement brownien},
     booktitle = {S\'eminaire Bourbaki : volume 2004/2005, expos\'es 938-951},
     author = {Collectif},
     series = {Ast\'erisque},
     note = {talk:951},
     pages = {469--480},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {307},
     year = {2006},
     zbl = {1126.60028},
     mrnumber = {2296427},
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     url = {http://archive.numdam.org/item/SB_2004-2005__47__469_0/}
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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://archive.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. | Article | MR 969362 | Zbl 0679.60013

[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. | Article | MR 143257 | Zbl 0121.13003

[3] O. Daviaud - “ Extremes of the discrete two-dimensional Gaussian free field”, Ann. Probab. 34 (2006), no. 3, p. 962-986. | MR 2243875 | Zbl 1104.60062

[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 2228383 | Zbl 1102.60009

[5] A. Dembo, Y. Peres & J. Rosen - “How large a disc is covered by a random walk in n steps ?”, prépublication. | Article | Zbl 1123.60026

[6] -, “Brownian motion on compact manifolds : cover time and late points”, Electron. J. Probab. 8 (2003), no. 15, 14 pp. (electronic). | EuDML 124682 | MR 1998762 | Zbl 1063.58021

[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 2206347 | Zbl 1100.60057

[8] -, “Thick points for transient symmetric stable processes”, Electron. J. Probab. 4 (1999), no. 10, 13 pp. (electronic). | EuDML 120048 | MR 1690314 | Zbl 0927.60077

[9] -, “Thick points for spatial Brownian motion : multifractal analysis of occupation measure”, Ann. Probab. 28 (2000), no. 1, p. 1-35. | MR 1755996 | Zbl 1130.60311

[10] -, “Thin points for Brownian motion”, Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 6, p. 749-774. | EuDML 77678 | Numdam | MR 1797392 | Zbl 0977.60073

[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 1846031 | Zbl 1008.60063

[12] -, “Thick points for intersections of planar sample paths”, Trans. Amer. Math. Soc. 354 (2002), no. 12, p. 4969-5003. | MR 1926845 | Zbl 1007.60077

[13] -, “Cover times for Brownian motion and random walks in two dimensions”, Ann. of Math. (2) 160 (2004), no. 2, p. 433-464. | MR 2123929 | Zbl 1068.60018

[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 996984 | Zbl 0676.60038

[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 790875 | Zbl 0593.60072

[16] -, “Three problems on the random walk in Z d , Studia Sci. Math. Hungar. 26 (1991), p. 309-320. | MR 1180496 | Zbl 0774.60036

[17] P. Erdős & S. J. Taylor - “Some problems on the structure of random walk paths”, Acta Math. Sci. Hungar. 11 (1960), p. 137-162. | Article | MR 121870 | Zbl 0091.13303

[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 402883 | Zbl 0307.60045

[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 1278083 | Zbl 0789.60019

[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 1229519 | Zbl 0779.60068

[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 912654 | Zbl 0613.60071

[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. | Article | MR 145599 | Zbl 0119.14602

[23] P. Révész - “Clusters of a random walk on the plane”, Ann. Probab. 21 (1993), no. 1, p. 318-328. | MR 1207228 | Zbl 0770.60034

[24] -, “Covering problems”, Theory Probab. Appl. 38 (1993), p. 367-379. | MR 1317989 | Zbl 0807.60068

[25] -, Random walk in random and non-random environments, second 'ed., World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. | MR 2168855 | Zbl 1090.60001

[26] J. Rosen - “A random walk proof of the Erdős-Taylor conjecture”, Period. Math. Hungar. 50 (2005), no. 1-2, p. 223-245. | MR 2162811 | Zbl 1098.60045

[27] S. J. Taylor - “Regularity of irregularities on a Brownian path”, Ann. Inst. Fourier 24 (1974), p. 195-203. | Article | EuDML 74172 | Numdam | MR 410959 | Zbl 0262.60059

[28] B. Tóth - “No more than three favorite sites for simple random walk”, Ann. Probab. 29 (2001), no. 1, p. 484-503. | MR 1825161 | Zbl 1031.60036

[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 2079672 | Zbl 1057.60078