On mean curvature functions of Brownian paths

We consider the path Zt described by a standard Brownian motion in on some time interval [0,t]. This is a random compact subset of . Using the support (curvature) measures of [D. Hug, G. Last, W. Weil, A local Steiner-type formula for general closed sets and applications, Math. Z. 246 (2004) 237-272] we introduce and study two mean curvature functions of Brownian motion. The geometric interpretation of these functions can be based on the Wiener sausage of radius r>0 which is the set of all points whose Euclidean distance d(Zt,x) from Zt is at most r. The mean curvature functions can be easily expressed in terms of the Gauss and mean curvature of as integrated over the positive boundary of . We will show that these are continuous functions of locally bounded variation. A consequence is that the volume of is almost surely differentiable at any fixed r>0 with the derivative given as the content of the positive boundary of . This will imply that also the expected volume of is differentiable with the derivative given as the expected content of the positive boundary of . In fact it has been recently shown in [J. Rataj, V. Schmidt, E. Spodarev, On the expected surface area of the Wiener sausage (2005) (submitted for publication) http://www.mathematik.uni-ulm.de/stochastik/] that for d =4 this is true at least for almost all r>0. The paper [J. Rataj, V. Schmidt, E. Spodarev, On the expected surface area of the Wiener sausage (2005) (submitted for publication) http://www.mathematik.uni-ulm.de/stochastik/] then proceeds to use a result from [ A.M. Berezhkovskii, Yu.A. Makhnovskii, R.A. Suris, Wiener sausage volume moments, J. Stat. Phys. 57 (1989) 333-346] to get explicit formulae for this expected surface content. We will use here this result to derive a linear constraint on the mean curvature functions. For d=3 we will provide a more detailed analysis of the mean curvature functions based on a classical formula in [F. Spitzer, Electrostatic capacity, heat flow, and Brownian motion, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964) 110-121].