On mean curvature functions of Brownian paths

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:116

Article

Last, Guenter

关键词: Brownian motion;    Wiener sausage;    curvature;    support measures;   

DOI  :  10.1016/j.spa.2006.05.003

来源: Elsevier

【 摘 要 】

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 Rd. 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(circle plus r)(t), of radius r > 0 which is the set of all points x is an element of 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(circle plus r)(t), as integrated over the positive boundary of Z(circle plus r)(t). We will show that these are continuous functions of locally bounded variation. A consequence is that the volume of Z(circle plus r)(t), is almost surely differentiable at any fixed r > 0 with the derivative given as the content of the positive boundary of Z(circle plus r)(t). This will imply that also the expected volume of Z(circle plus r)(t) is differentiable with the derivative given as the expected content of the positive boundary of ZEt)r, 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(circle plus 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) 333-3461 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]. (c) 2006 Elsevier B.V. All rights reserved.

【 授权许可】

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