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

Itai Benjamini; Nathanael Berestycki

doi:10.1214/10-AIHP371

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

Itai Benjamini, Nathanael Berestycki

Department of Mathematics

Weizmann Institute of Science

Research output: Contribution to journal › Article › peer-review

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 L _s le; f (s),for all s≤ t, where L_s 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^-3/2 f(t) is integrable. We conjecture that this condition is sharp and present a number of open problems.

Original language	English
Pages (from-to)	539-558
Number of pages	20
Journal	Annales De L Institut Henri Poincare-Probabilites Et Statistiques
Volume	47
Issue number	2
DOIs	https://doi.org/10.1214/10-AIHP371
State	Published - May 2011

All Science Journal Classification (ASJC) codes

Statistics and Probability
Statistics, Probability and Uncertainty

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10.1214/10-AIHP371

Cite this

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title = "An integral test for the transience of a Brownian path with limited local time",

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 L s le; 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-3/2 f(t) is integrable. We conjecture that this condition is sharp and present a number of open problems.",

author = "Itai Benjamini and Nathanael Berestycki",

note = "PIMS; UBCWe thank Ross Pinsky for some detailed comments improving on a first version of this paper, and Marc Yor for showing us a draft of [13]. N. B. would like to gratefully acknowledge PIMS and UBC, whose support was determinant for the completion of this work, as well as the hospitality of the Weizmann Institute. We would like to thank a first anonymous referee, whose careful reading made us reformulate our results after finding some mistakes in an earlier version.",

year = "2011",

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doi = "10.1214/10-AIHP371",

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AU - Berestycki, Nathanael

N1 - PIMS; UBCWe thank Ross Pinsky for some detailed comments improving on a first version of this paper, and Marc Yor for showing us a draft of [13]. N. B. would like to gratefully acknowledge PIMS and UBC, whose support was determinant for the completion of this work, as well as the hospitality of the Weizmann Institute. We would like to thank a first anonymous referee, whose careful reading made us reformulate our results after finding some mistakes in an earlier version.

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N2 - 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 L s le; 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-3/2 f(t) is integrable. We conjecture that this condition is sharp and present a number of open problems.

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M3 - مقالة

SN - 0246-0203

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EP - 558

JO - Annales De L Institut Henri Poincare-Probabilites Et Statistiques

JF - Annales De L Institut Henri Poincare-Probabilites Et Statistiques

IS - 2

ER -

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

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