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Pré-Publication, Document De Travail Année : 2022

The *-Vertex-Reinforced Jump Process

Résumé

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.

Dates et versions

hal-03795095 , version 1 (03-10-2022)

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Christophe Sabot, Pierre Tarrès. The *-Vertex-Reinforced Jump Process. 2022. ⟨hal-03795095⟩
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