The probability that Brownian motion almost contains a line (Q1359047)

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.