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), consider the measures mut 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 mut are tight, and that any weak limit of mut as t tends to infinity is transient provided that t-3/2f(t) is integrable. We conjecture that this condition is sharp and present a number of open problems.
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