Open Access
2022 Rate of escape of conditioned Brownian motion
Orphée Collin, Francis Comets
Author Affiliations +
Electron. J. Probab. 27: 1-26 (2022). DOI: 10.1214/21-EJP737


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:

  • we find an integral test on the function g so that the future minima process drops below the barrier exp{lnt×g(lnlnt)} at arbitrary large times;

  • we show that the future minima process exceeds Kt×lnlnlntat 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.

Funding Statement

Francis Comets is partially supported by ANR SWIWS.


Download Citation

Orphée Collin. Francis Comets. "Rate of escape of conditioned Brownian motion." Electron. J. Probab. 27 1 - 26, 2022.


Received: 4 April 2021; Accepted: 30 December 2021; Published: 2022
First available in Project Euclid: 2 March 2022

arXiv: 2102.09636
MathSciNet: MR4387839
Digital Object Identifier: 10.1214/21-EJP737

Primary: 60G17 , 60J60 , 60J65 , 60K35

Keywords: autoregressive process , Bessel process , Brownian motion , Conditioning , random difference equation , regeneration , transience , upper-class and lower-class , Wiener moustache

Vol.27 • 2022
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