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.
Publié le : 2011-05-15
Classification:
Brownian motion,
Conditioning,
Local time,
Entropic repulsion,
Integral test,
Transience,
Recurrence,
60G17,
60J65,
60K37
@article{1300887281,
author = {Benjamini, Itai and Berestycki, Nathana\"el},
title = {An integral test for the transience of a Brownian path with limited local time},
journal = {Ann. Inst. H. Poincar\'e Probab. Statist.},
volume = {47},
number = {1},
year = {2011},
pages = { 539-558},
language = {en},
url = {http://dml.mathdoc.fr/item/1300887281}
}
Benjamini, Itai; Berestycki, Nathanaël. An integral test for the transience of a Brownian path with limited local time. Ann. Inst. H. Poincaré Probab. Statist., Tome 47 (2011) no. 1, pp. 539-558. http://gdmltest.u-ga.fr/item/1300887281/