A continuous random operator associated with the Vertex Reinforced Jump Process on the circle and the real line
Abstract
In this paper, we focus on the scaling-limit of the random potential associated with the Vertex Reinforced Jump Process (VRJP) on one-dimensional graphs. Moreover, we give a few applications of this scaling-limit. By considering a relevant scaling of , we contruct a continuous-space version of the random Schr{\"o}dinger operator which is associated with the VRJP on circles and on R. We also compute the integrated density of states of this operator on R which has a remarkably simple form. Moreover, by means of the same scaling, we obtain a new proof of the Matsumoto-Yor properties concerning the geometric Brownian motion which were proved in [MY01]. This new proof is based on some fundamental properties of the random potential . We use also the scaling-limit of in order to prove new identities in law involving exponential functionals of the Brownian motion which generalize the Dufresne identity.
Keywords
Cite
@article{arxiv.2308.01120,
title = {A continuous random operator associated with the Vertex Reinforced Jump Process on the circle and the real line},
author = {V Rapenne and C Sabot},
journal= {arXiv preprint arXiv:2308.01120},
year = {2023}
}