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

Itai Benjamini; Nathanaël Berestycki

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 2, page 539-558
  • ISSN: 0246-0203

Abstract

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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 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.

How to cite

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Benjamini, Itai, and 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 47.2 (2011): 539-558. <http://eudml.org/doc/242832>.

@article{Benjamini2011,
abstract = {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.},
author = {Benjamini, Itai, Berestycki, Nathanaël},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {brownian motion; conditioning; local time; entropic repulsion; integral test; transience; recurrence; Brownian motion},
language = {eng},
number = {2},
pages = {539-558},
publisher = {Gauthier-Villars},
title = {An integral test for the transience of a brownian path with limited local time},
url = {http://eudml.org/doc/242832},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Benjamini, Itai
AU - Berestycki, Nathanaël
TI - An integral test for the transience of a brownian path with limited local time
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 2
SP - 539
EP - 558
AB - 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.
LA - eng
KW - brownian motion; conditioning; local time; entropic repulsion; integral test; transience; recurrence; Brownian motion
UR - http://eudml.org/doc/242832
ER -

References

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  11. [11] L. C. Rogers and D. Williams. Diffusions, Markov Processes and Martingales, Vol. 2, 2nd edition. Cambridge Univ. Press, Cambridge, 2000. Zbl0949.60003MR1780932
  12. [12] B. Roynette, P. Vallois and M. Yor. Some penalisations of the Wiener measure. Japan. J. Math. 1 (2006) 263–290. Zbl1160.60315MR2261065
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