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

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.

Étant donnée une fonction croissante 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.

DOI : https://doi.org/10.1214/10-AIHP371
Classification:  60G17,  60J65,  60K37
Keywords: brownian motion, conditioning, local time, entropic repulsion, integral test, transience, recurrence
@article{AIHPB_2011__47_2_539_0,
     author = {Benjamini, Itai and Berestycki, Nathana\"el},
     title = {An integral test for the transience of a brownian path with limited local time},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {2},
     year = {2011},
     pages = {539-558},
     doi = {10.1214/10-AIHP371},
     zbl = {1216.60028},
     mrnumber = {2814422},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2011__47_2_539_0}
}
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, Volume 47 (2011) no. 2, pp. 539-558. doi : 10.1214/10-AIHP371. http://www.numdam.org/item/AIHPB_2011__47_2_539_0/

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