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

Mikhail A. Lifshits, Zhan Shi

• 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.