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.

Classification:  60G17,  60J65,  60K37
Keywords: brownian motion, conditioning, local time, entropic repulsion, integral test, transience, recurrence
     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 = {}
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.

[1] M. Barlow and E. Perkins. Brownian motion at a slow point. Trans. Amer. Math. Soc. 296 (1986) 741-775. | MR 846605 | Zbl 0602.60041

[2] I. Benjamini and N. Berestycki. Random paths with bounded local time. J. Eur. Math. Soc. 12 (2010) 819-854. | MR 2654081 | Zbl 1202.60131

[3] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley, New York, 1999. | MR 1700749 | Zbl 0172.21201

[4] R. Durrett. Probability: Theory and Examples, 3rd edition. Duxbury Press, Belmont, CA, 2004. | MR 2722836 | Zbl 0709.60002

[5] W. Feller. An Introduction to Probability Theory and its Applications, Vol. II, 2nd edition. Wiley, New York, 1971. | MR 270403 | Zbl 0138.10207

[6] R. Van Der Hofstad and W. König. A survey of one-dimensional polymers. J. Stat. Phys. 103 (2001) 915-944. | MR 1851362 | Zbl 1126.82313

[7] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North-Holland-Kodansha, Amsterdam and Tokyo, 1981. | MR 637061 | Zbl 0684.60040

[8] J. Najnudel. Construction of an Edwards' probability measure on C(ℝ+, ℝ). Ann. Probab. (2010). To appear. Preprint. Available at arXiv:0801.2751. | MR 2683631 | Zbl pre05817049

[9] J. Pitman. The SDE solved by local times of a Brownian excursion or bridge derived from height profile of a random tree or forest. Ann. Probab. 27 (1999) 261-283. | MR 1681110 | Zbl 0954.60060

[10] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin, 1999. | MR 1725357 | Zbl 0917.60006

[11] L. C. Rogers and D. Williams. Diffusions, Markov Processes and Martingales, Vol. 2, 2nd edition. Cambridge Univ. Press, Cambridge, 2000. | MR 1780932 | Zbl 0949.60003

[12] B. Roynette, P. Vallois and M. Yor. Some penalisations of the Wiener measure. Japan. J. Math. 1 (2006) 263-290. | MR 2261065 | Zbl 1160.60315

[13] B. Roynette and M. Yor. Penalising Brownian Paths: Rigorous Results and Meta-Theorems. Lecture Notes in Math. 1969. Springer, Berlin, 2009. | MR 2504013 | Zbl 1190.60002

[14] M. Yor. Local Times and Excursions for Brownian Motion: A Concise Introduction. Lecciones en Matematicas 1. Universidad Central de Venezuela, Caracas, 1995.