Continuous time vertex-reinforced jump processes

Abstract

We study the continuous time integer valued process Xt, t ≥ 0, which jumps to each of its two nearest neighbors at the rate of one plus the total time the process has previously spent at that neighbor. We show that the proportion of the time before t which this process spends at integers j converges to positive random variables Vj, which sum to one, and whose joint distribution is explicitly described. We also show limt→∞ max0≤s≤t Xs/log t = 2.768...