On an upper bound of the Euler characteristic of the Wiener sausage
Let \(B[0,1]\) denote the image of the standard planar Brownian motion run for unit time, and for \(\gamma > 0\) write \(S_\gamma [0,1] = \{ x \in \mathbb{R}^2 : d (x , B[0,1] ) \leq \gamma \}\) for the \(\gamma\)-radius Wiener sausage, where \(d (x, A) = \inf_{y \in A} \| x - y\|\). For \(u \geq 0\), let \(N_\gamma [u, \infty)\) be the number of connected components of \(\mathbb{R}^2 \setminus S_\gamma [0,1]\) whose area is at least \(u\). The author proves that, a.s., \[ N_\gamma [ u , \infty) \sim \frac{2\pi}{u (\log u)^2} \] uniformly as \(u \to 0\) with \(0 < \gamma < (u /\pi)^b\), for any \(b > 1/2\). Considering the Euler characteristic \(\chi (\gamma)\) of \(S_\gamma [0,1]\), this result, together with the fact that \(\chi (\gamma) = 1 - N_\gamma [ 0, \infty) \leq 1 - N_\gamma [u, \infty)\) for any \(u \geq 0\), yields an asymptotic upper bound on \(\chi(\gamma)\): for example, for any \(p < 2\), a.s., \(\chi (\gamma) \leq 1 - \gamma^{-p}\) for all \(\gamma\) sufficiently small. It was already known that \(\mathbb{E} | \chi(\gamma) |\) is finite for any \(\gamma >0\) [\textit{J. Rataj} et al., Math. Nachr. 282, No. 4, 591--603 (2009; Zbl 1166.60049)].
scientific article; zbMATH DE number 6377116
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