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May 2020 Hitting times of interacting drifted Brownian motions and the vertex reinforced jump process
Christophe Sabot, Xiaolin Zeng
Ann. Probab. 48(3): 1057-1085 (May 2020). DOI: 10.1214/19-AOP1381


Consider a negatively drifted one-dimensional Brownian motion starting at positive initial position, its first hitting time to 0 has the inverse Gaussian law. Moreover, conditionally on this hitting time, the Brownian motion up to that time has the law of a three-dimensional Bessel bridge. In this paper, we give a generalization of this result to a family of Brownian motions with interacting drifts, indexed by the vertices of a conductance network. The hitting times are equal in law to the inverse of a random potential that appears in the analysis of a self-interacting process called the vertex reinforced jump process (Ann. Probab. 45 (2017) 3967–3986; J. Amer. Math. Soc. 32 (2019) 311–349). These Brownian motions with interacting drifts have remarkable properties with respect to restriction and conditioning, showing hidden Markov properties. This family of processes are closely related to the martingale that plays a crucial role in the analysis of the vertex reinforced jump process and edge reinforced random walk (J. Amer. Math. Soc. 32 (2019) 311–349) on infinite graphs.


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Christophe Sabot. Xiaolin Zeng. "Hitting times of interacting drifted Brownian motions and the vertex reinforced jump process." Ann. Probab. 48 (3) 1057 - 1085, May 2020.


Received: 1 May 2018; Revised: 1 June 2019; Published: May 2020
First available in Project Euclid: 17 June 2020

zbMATH: 07226354
MathSciNet: MR4112708
Digital Object Identifier: 10.1214/19-AOP1381

Primary: 60J65, 60K35, 60K37
Secondary: 60J60, 81T25, 81T60, 82B44

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.48 • No. 3 • May 2020
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