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

TL;DR

This paper investigates the conditions under which a one-dimensional Brownian motion with a self-repelling constraint on local time becomes transient, establishing a tightness result and a criterion involving the integrability of a scaled function.

Contribution

It introduces a new integral test for the transience of Brownian paths conditioned on limited local time, extending understanding of self-repelling Brownian motions.

Findings

Measures are tight under the given conditioning.

Transience occurs if t^{-3/2}f(t) is integrable.

Open problems and conjectures are proposed.

Abstract

We study a one-dimensional Brownian motion conditioned on a self-repelling behaviour. Given a nondecreasing positive function f(t), consider the measures mu_t obtained by conditioning a Brownian path so that L_s< 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 mu_t are tight, and that any weak limit of mu_t as t tends to infinity 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.