Reinforced random walk

Abstract

Let ai, i≧1, be a sequence of nonnegative numbers. Difine a nearest neighbor random motion {Mathematical expression}=X0, X1, ... on the integers as follows. Initially the weight of each interval (i, i+1), i an integer, equals 1. If at time n an interval (i, i+1) has been crossed exactly k times by the motion, its weight is {Mathematical expression}. Given (X0, X1, ..., Xn)=(i0, i1, ..., in), the probability that Xn+1 is in-1 or in+1 is proportional to the weights at time n of the intervals (in-1, in) and (in, iin+1). We prove that {Mathematical expression} either visits all integers infinitely often a.s. or visits a finite number of integers, eventually oscillating between two adjacent integers, a.s., and that {Mathematical expression}Xn/n=0 a.s. For much more general reinforcement schemes we prove P ( {Mathematical expression} visits all integers infinitely often)+P ( {Mathematical expression} has finite range)=1. © 1990 Springer-Verlag.