Annals of Probability

Hitting times of interacting drifted Brownian motions and the vertex reinforced jump process

Christophe Sabot and Xiaolin Zeng

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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.

Article information

Ann. Probab., Volume 48, Number 3 (2020), 1057-1085.

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

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Mathematical Reviews number (MathSciNet)

Primary: 60J65: Brownian motion [See also 58J65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments
Secondary: 60J60: Diffusion processes [See also 58J65] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 81T25: Quantum field theory on lattices 81T60: Supersymmetric field theories

Inverse Gaussian law hitting time of Brownian motion self-interacting processes vertex reinforced jump process random Schrödinger operator


Sabot, Christophe; Zeng, Xiaolin. Hitting times of interacting drifted Brownian motions and the vertex reinforced jump process. Ann. Probab. 48 (2020), no. 3, 1057--1085. doi:10.1214/19-AOP1381.

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