TY - JOUR

T1 - Hitting times of interacting drifted brownian motions and the vertex reinforced jump process

AU - Sabot, Christophe

AU - Zeng, Xiaolin

N1 - Publisher Copyright:
© Institute of Mathematical Statistics, 2020.

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - Hitting time of brownian motion

KW - Inverse Gaussian law

KW - Random Schrödinger operator

KW - Self-interacting processes

KW - Vertex reinforced jump process

UR - http://www.scopus.com/inward/record.url?scp=85089235157&partnerID=8YFLogxK

U2 - 10.1214/19-AOP1381

DO - 10.1214/19-AOP1381

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AN - SCOPUS:85089235157

SN - 0091-1798

VL - 48

SP - 1057

EP - 1085

JO - Annals of Probability

JF - Annals of Probability

IS - 3

ER -