Rate of escape of conditioned Brownian motion

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.