Rate of escape of conditioned Brownian motion

TL;DR

This paper investigates the escape rate of a two-dimensional Brownian motion conditioned to stay outside the unit disk, providing sharp criteria for its future minima and long-term behavior using renewal structures.

Contribution

It introduces a detailed analysis of the conditioned Brownian motion's escape rate, including integral tests and probabilistic thresholds, with new renewal techniques.

Findings

Integral test for the process dropping beyond certain barriers

Thresholds for the future minima exceeding specific functions

Sharp results on the long-term norm behavior

Abstract

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.