  We introduce a non-reversible generalization of the Vertex-Reinforced Jump
Process (VRJP), which we call *-Vertex-Reinforced Jump Process (*-VRJP). It can
be seen as the continuous-time counterpart of the *-Edge-Reinforced Random Walk
(*-ERRW) (see [2,4]), which is itself a non-reversible generalization of the
original ERRW introduced by Coppersmith and Diaconis in 1986. In contrast to
the classical VRJP, the *-VRJP is not exchangeable after time-change, but we
show that with some appropriate randomization of the initial local time, it
becomes partially exchangeable after time-change. We provide a representation
of the "randomized" *-VRJP as mixture of Markov jump processes with an explicit
mixing measure, and show that the "non-randomized" *-VRJP can be written as a
mixture of conditioned Markov processes. We also give a counterpart of the
$\beta$-random potential introduced in [25] for the VRJP, which leads to new
identities between integrals.
