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The probability that Brownian motion almost contains a line. (English) Zbl 0880.60040
Let $$\{B_t, t\geq 0\}$$ be a two-dimensional Brownian motion and let $${\mathcal R}= B[0,1]$$ be the range of its trajectory up to time 1. Denote by $${\mathcal R}_\varepsilon= \{x\in\mathbb{R}^2:|y-x|<\varepsilon$$ for some $$y\in{\mathcal R}\}$$ the Wiener sausage with radius $$\varepsilon$$. Consider $$P_\varepsilon= P({\mathcal R}_\varepsilon\supset[0, 1]\times\{0\})$$, the probability that the Wiener sausage contains the unit interval on the $$x$$-axis. It is shown that for sufficiently small $$\varepsilon$$, $-c_4\log^4\varepsilon\leq \log P_\varepsilon\leq c_1- c_2\log^2\varepsilon/\log^2|\log\varepsilon|$ for some positive $$c_1$$, $$c_2$$, $$c_4$$. It follows that a two-dimensional Brownian motion, run for infinite time, almost surely contains no line segment.

##### MSC:
 60G17 Sample path properties 60J65 Brownian motion
##### Keywords:
planar Brownian motion; range of trajectory; Wiener sausage
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