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: 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 K t×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.
Highlights
This paper is devoted to the planar Brownian motion conditioned to stay outside the unit ball B(0, 1) at all times
The Wiener moustache is obtained by gluing two instances of planar Brownian motion conditioned to stay outside the unit ball, which are independent except that they share the same starting point
Consider W a two-dimensional standard Brownian motion and denote by Px the law of W starting at x, W a Brownian motion conditioned to stay outside the unit ball, and denote by Px its law when starting at x, and R = |W | its Euclidean norm with Pr the corresponding law (r = |x|)
Summary
This paper is devoted to the planar Brownian motion conditioned to stay outside the unit ball B(0, 1) at all times. The two-dimensional Brownian motion is critically recurrent, but conditioning it outside the unit ball turns it into (delicately) transient. Which is non-decreasing to ∞ a.s. The corresponding model in the discrete case, the two-dimensional simple random walk conditioned to avoid the origin at all times, has motivated many recent papers. The Wiener moustache is obtained by gluing two instances (for positive and negative times, see Figure 1 in [9]) of planar Brownian motion conditioned to stay outside the unit ball, which are independent except that they share the same starting point (see Lemma 3.9 in [9]). We prove some results showing that R somewhat behaves at large times like the two-dimensional Bessel process.
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