Problèmes de recouvrement et points exceptionnels pour la marche aléatoire et le mouvement brownien

Shi, Zhan

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.

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