  We study the norm of the two-dimensional Brownian motion conditioned to stay
outside the unit disk at all times. By conditioning the process is changed from
barely recurrent to slightly transient. We obtain sharp results on the rate of
escape to infinity of the process of future minima: (i) we find an integral
test on the function $g$ so that the future minima process drops beyond the
barrier $\exp \{ \ln t \times g(\ln \ln t)\}$ at arbitrary large times; (ii) we
show that the future minima process exceeds $K \sqrt{ t \times \ln \ln \ln t}$
at arbitrary large times with probability 0 [resp., 1] if $K$ is larger [resp.,
smaller] than some positive constant. For this, we introduce a renewal
structure attached to record times and values. Additional results are given for
the long time behavior of the norm.
