Continuous time vertex-reinforced jump processes
The aim of this article is the study of a vertex-reinforced jump process conceived by W. Werner. This is a continuous time right-continuous nearest neighbour random walk \(X\) on the integers which jumps to a (nearest) neighbour at the rate of one plus the total time spent at that neighbour previously. The authors show that the portion of time before \(t\) the process spends at some integer \(i\) converges for \(t\rightarrow\infty\) to a positive random variable \(V_i\). The sum of \(V_i\) over the integers is one and their joint distribution is explicitly described. Moreover, they prove that \(\lim_{t\rightarrow\infty}\max_{0\leq s\leq t}X_s/\log t\approx 2.77\). The symmetry of the law of \(X\) gives the analog for the minimum. As a consequence \(X\) is recurrent. The study of vertex-reinforced jump processes is motivated by the studies on bond-reinforced random walks by Coppersmith and Diaconis and on vertex-reinforced random walks by \textit{R. Pemantle} [Ann. Probab. 16, No. 3, 1229-1241 (1988; Zbl 0648.60077)] and \textit{R. Pemantle} and \textit{S. Volkov} [ibid. 27, No. 3, 1368-1388 (1999; Zbl 0960.60041)].
scientific article; zbMATH DE number 1836958