On the Expected Surface Area of the Wiener Sausage

There exists an extensive literature on Wiener sausages; see e.g. [25] and the references therein. Nevertheless, relatively little is known so far about the geometry of Wiener sausages due to the complex nature of their realizations. One possible description of the geometric structure of (sufficiently regular) subsets of R is given by their d + 1 intrinsic volumes or Minkowski functionals including the usual volume, surface area and other curvature measures as well as the Euler–Poincaré characteristic; see e.g. [5]. As far as it is known to the authors, the only Minkowski functional of Wiener sausages studied in the literature is their expected d–dimensional Lebesgue measure. Thus, explicit formulae for the expected volume of Wiener sausages and its asymptotic behavior for T → 0 or T → ∞ can be found in [9], [13], [23]; see also [2]. The results of corresponding simulation studies in three dimensions are discussed in [31]. Further limit theorems and deviation results for the volume of Wiener sausages are proved in [15] and [27]. Asymptotic long–time behavior of its moment generating function is given in [4], [26], [28]; see also [25], pp. 201 and 315.