Details

Autor(en) / Beteiligte

• Last, Günter

Titel

On mean curvature functions of Brownian paths

Ist Teil von

• Stochastic processes and their applications, 2006-12, Vol.116 (12), p.1876-1891

Ort / Verlag

Amsterdam: Elsevier B.V

Erscheinungsjahr

2006

Link zum Volltext

Quelle

EZB Electronic Journals Library

Beschreibungen/Notizen

• We consider the path Z t described by a standard Brownian motion in R d on some time interval [ 0 , t ] . This is a random compact subset of R d . 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 Z ⊕ r t of radius r > 0 which is the set of all points x ∈ R d whose Euclidean distance d ( Z t , x ) from Z t is at most r . The mean curvature functions can be easily expressed in terms of the Gauss and mean curvature of Z ⊕ r t as integrated over the positive boundary of Z ⊕ r t . We will show that these are continuous functions of locally bounded variation. A consequence is that the volume of Z ⊕ r t is almost surely differentiable at any fixed r > 0 with the derivative given as the content of the positive boundary of Z ⊕ r t . This will imply that also the expected volume of Z ⊕ r t is differentiable with the derivative given as the expected content of the positive boundary of Z ⊕ r t . 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 ≤ 3 the derivative is just the expected surface content of Z ⊕ r t and 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].