An integral test for the transience of a brownian path with limited local time
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 2, pp. 539-558.

Étant donnée une fonction croissante f(t), t≥0, considérons la mesure μt obtenue lorsqu'on on conditionne un mouvement brownien de sorte que Ls≤f(s), pour tout s≤t, 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.

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 Ls≤f(s), for all s≤t, 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.

DOI : 10.1214/10-AIHP371
Classification : 60G17, 60J65, 60K37
Mots-clés : brownian motion, conditioning, local time, entropic repulsion, integral test, transience, recurrence 
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Benjamini, Itai; Berestycki, Nathanaël. An integral test for the transience of a brownian path with limited local time. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 2, pp. 539-558. doi : 10.1214/10-AIHP371. http://www.numdam.org/articles/10.1214/10-AIHP371/

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