  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 $d\geq 3$).
