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

Abstract

This paper concerns the Vertex Reinforced Jump Process (VRJP) and its representations as a Markov process in random environment. In [21], it was shown that the VRJP on finite graphs, under a certain time rescaling, has the distribution of a mixture of Markov jump processes. This representation was extended to infinite graphs in [23], by introducing a random potential $\beta $. In this paper, we show that all possible representations of the VRJP as a mixture of Markov processes can be expressed in a similar form as in [23], using the random field $\beta $ and harmonic functions for an associated operator $H_\beta $. 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 $d\geq 3$).