# Representations of the Vertex Reinforced Jump Process as a mixture of Markov processes on $\mathbb{Z}^d$ and infinite trees

## TL;DR

This paper investigates the Vertex Reinforced Jump Process (VRJP) and demonstrates its unique representation as a mixture of Markov processes on $
abla^d$ and infinite trees, revealing different behaviors in various structures.

## Contribution

It provides a unified framework for representing VRJP as a mixture of Markov processes and establishes uniqueness on $
abla^d$, while constructing multiple representations on infinite trees.

## Key findings

- VRJP has a unique representation on $
abla^d$ due to trivial Martin boundary.
- Multiple distinct representations of VRJP exist on infinite $d$-regular trees when transient.
- The paper links VRJP representations to harmonic functions and random potentials.

## Abstract

This paper concerns the Vertex Reinforced Jump Process (VRJP) and its representations as a Markov process in random environment. We show that all possible representations of the VRJP as a mixture of Markov processes can be expressed in a similar form, using a random potential and harmonic functions for an associated operator. This allows to show that the VRJP on $\mathbb{Z}^d$ (with certain initial conditions) has a unique representation, by proving that an associated Martin boundary is trivial. Moreover, on infinite trees, we construct a family of representations, that are all different when the VRJP is transient and the tree is $d$-regular (with $d\geq 3$).