Stochastic processes with competing reinforcements

Abstract

We introduce a simple but powerful strategy to study processes driven by two or more reinforcement mechanisms in competition. We apply our method to two types of models: to nonconservative zero range processes on finite graphs, and to multi-particle random walks with positive and negative reinforcement on the edges. The results hold for a broad class of reinforcement functions, including those with superlinear growth. Our strategy consists in a comparison of the original processes with suitable reference models. To implement the comparison we estimate an object reminiscent to the Radon–Nikodym derivative on a carefully chosen set of trajectories. Our results describe the almost sure long time behaviour of the processes. We also prove a phase transition depending on the strength of the reinforcement functions.