The escape rate of favorite sites of simple random walk and Brownian motion

• Mikhail A. Lifshits ^[2] ; Zhan Shi ^[1]
  1. [1] Pierre and Marie Curie University

     Pierre and Marie Curie University

     París, Francia

     
  2. [2] St. Petersburg State University
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 32, Nº. 1, 1, 2004, págs. 129-152
• Idioma: inglés
• DOI: 10.1214/aop/1078415831
• Texto completo no disponible (Saber más ...)
• Resumen

  • Consider a simple symmetric random walk on the integer lattice \ZB. For each n, let V(n) denote a favorite site (or most visited site) of the random walk in the first n steps. A somewhat surprising theorem of Bass and Griffin [Z. Wahrsch. Verw. Gebiete 70 (1985) 417--436] says that V is almost surely transient, thus disproving a previous conjecture of Erdős and Révész [Mathematical Structures--Computational Mathematics--Mathematical Modeling 2 (1984) 152--157]. More precisely, Bass and Griffin proved that almost surely, lim infn→∞|V(n)|n1/2(logn)−γ equals 0 if γ<:1, and is infinity if γ>11 (eleven). The present paper studies the rate of escape of V(n). We show that almost surely, the "lim\,inf'' expression in question is 0 if γ≤1, and is infinity otherwise. The corresponding problem for Brownian motion is also studied.