HITTING TIMES OF INTERACTING DRIFTED BROWNIAN MOTIONS AND THE VERTEX REINFORCED JUMP PROCESS

• 成果类型: Article
• 署名作者: Sabot, Christophe; Zeng, Xiaolin
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; Universites de Strasbourg Etablissements Associes; Universite de Strasbourg
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1381
• 发表日期: 2020
• 页码: 1057-1085
• 关键词: exit time

摘要

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.