The *-Vertex-Reinforced Jump Process
Sabot, Christophe; Tarres, Pierre (2022), The *-Vertex-Reinforced Jump Process. https://basepub.dauphine.psl.eu/handle/123456789/23326
TypeDocument de travail / Working paper
Series titleCahier de recherche CEREMADE, Université Paris Dauphine-PSL
MetadataShow full item record
Institut Camille Jordan [ICJ]
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)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 β-random potential introduced in  for the VRJP, which leads to new identities between integrals.
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