The ⋆-vertex-reinforced jump process

We investigate the nonreversible generalization of the Vertex-Reinforced Jump Process (VRJP), called the ⋆-Vertex-Reinforced Jump Process (⋆-VRJP) and introduced in Bacallado, Sabot, Tarrès (2020). It can be seen as the continuous-time counterpart to the ⋆-Edge-Reinforced Random Walk (⋆-ERRW) (see Bacallado, Chodera, Pande (2009); Bacallado, Sabot, Tarrès (2020)), which is itself a nonreversible (and in fact Yaglom reversible) generalization of the original ERRW introduced by Coppersmith and Diaconis in 1986 (Coppersmith and Diaconis (1986)). In contrast to the classical VRJP, the ⋆-VRJP is not exchangeable after time-change, which leads to several difficulties and new phenomena. First, 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 a mixture of Yaglom reversible Markov jump processes with an explicit mixing measure, and we prove that the nonrandomized ⋆-VRJP can be written as a mixture of conditioned Markov processes. Second, we give a representation of the ⋆-VRJP in terms of a random Schrödinger operator. The corresponding representation for the classical VRJP has proved to be very useful in the understanding of its asymptotic behavior. The construction is based on several new and rather remarkable identities between integrals on the space of ⋆-symmetric and ⋆-antisymmetric functions on vertices. We give a description of the randomized ⋆-VRJP in terms of that random Schrödinger operator, which allows us to prove the representation of the randomized ⋆-VRJP as a mixture of Markov jump processes in a different and more analytic manner. Similar to the VRJP, we think that the representation by a random Schrödinger operator and the associated identities are key features of the ⋆-VRJP.