Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

An integral test for the transience of a Brownian path with limited local time

Itai Benjamini and Nathanaël Berestycki

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We study a one-dimensional Brownian motion conditioned on a self-repelling behaviour. Given a nondecreasing positive function f(t), t≥0, consider the measures μt obtained by conditioning a Brownian path so that Lsf(s), for all st, where Ls is the local time spent at the origin by time s. It is shown that the measures μt are tight, and that any weak limit of μt as t→∞ is transient provided that t−3/2f(t) is integrable. We conjecture that this condition is sharp and present a number of open problems.


Etant donnée une fonction croîssante f(t), t≥0, considérons la mesure μt obtenue lorsqu’on on conditionne un mouvement brownien de sorte que Lsf(s), pour tout st, où Ls est le temps local accumulé au temps s à l’origine. Nous montrons que les mesures μt sont tendues, et que toute limite faible de μt lorsque t→∞ est la loi d’un processus transient si t−3/2f(t) est intégrable. Nous conjecturons que cette condition est également nécessaire pour la transience et proposons un certain nombre de questions ouvertes.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 2 (2011), 539-558.

First available in Project Euclid: 23 March 2011

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Zentralblatt MATH identifier

Primary: 60G17: Sample path properties 60J65: Brownian motion [See also 58J65] 60K37: Processes in random environments

Brownian motion Conditioning Local time Entropic repulsion Integral test Transience Recurrence


Benjamini, Itai; Berestycki, Nathanaël. An integral test for the transience of a Brownian path with limited local time. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 2, 539--558. doi:10.1214/10-AIHP371.

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