Stochastic processes with competing reinforcements

TL;DR

This paper introduces a new technique to analyze stochastic processes driven by competing reinforcement mechanisms, applicable to models like zero range processes and multi-particle random walks, revealing phase transitions based on reinforcement strength.

Contribution

The paper develops a comparison-based method to study complex reinforcement-driven processes, extending analysis to superlinear reinforcement functions and identifying phase transitions.

Findings

Long-term behavior characterized for processes with competing reinforcements

Phase transition depends on reinforcement strength

Method applicable to broad class of reinforcement functions

Abstract

We introduce a simple but powerful technique to study processes driven by two or more reinforcement mechanisms in competition. We apply our method to two types of models: to non conservative 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 technique consists in a comparison of the original processes with suitable reference models. To implement the comparison we estimate a Radon-Nikodym derivative on a carefully chosen set of trajectories. Our results describe the almost surely long time behaviour of the processes. We also prove a phase transition depending on the strength of the reinforcement functions.