Vertex reinforced jump processes on one dimensional weighted environments

Reinforced random walks are processes whose future behavior is influenced by their history. A reinforced walk prefers to remain in areas it has already traversed rather than explore new territory. These processes can be broadly separated into two categories, edge reinforced processes or vertex reinforced processes, depending on the nature of the reinforcement. Two of these processes, Diaconis Walk, a discrete time edge reinforced model, and Vertex Reinforced Jump Process (VRJP), a continuous time vertex reinforced model, have been shown to exhibit considerable similarity of behavior in a range of different environments. This is despite substantial differences in definition, and disparate approaches and methods of proof. In one dimensional environments with general local bias, M. Takeshima proved that Diaconis Walk is transient iff the unreinforced walk on the same environment is transient. We prove an analogous result for VRJP