Recurrence and transience preservation for vertex reinforced jump processes in one dimension

Abstract

We show that the application of linear vertex reinforcement to one dimensional nearest neighbor Markov processes, yielding associated vertex reinforced jump processes, preserves both recurrence and transience. The analog for discrete time linear bond reinforcement is due to Takeshima. This together with another result we prove adds to the numerous known parallels between these two reinforcements. Martingales are the primary tool used to study vertex reinforced jump processes.