Journal of Statistical Physics, Vol. 57, Nos. l/2, 1989 Wiener Sausage Volume Moments A. M. Berezhkovskii, ~ Yu. A. Makhnovskii, 1 and R. A. Suris ~ Received November 23, 1988 The statistical characteristics of a spatial region visited by a spherical Brownian particle during time t (Wiener sausage) are investigated. The expectation value and dispersion of this quantity are obtained for a space of arbitrary dimension. In the one-dimensional case the distribution of probability density and the moments of any order are determined for this quantity. KEY WORDS: Brownian movement of spherical particle; Wiener sausage. 1. INTRODUCTION In the theory of random processes the spatial region visited by a spherical Brownian particle during time t is known as the Wiener sausage. (1'2) Its volume is important in an analysis of a number of physical processes. The average value of this random quantity was calculated for the first time in a pioneering work (3) for the two-dimensional case. In the present study we investigate the statistical characteristics of the Wiener sausage volume in a space of arbitrary dimension. The results are described in the following order~ Sections 2 and 3 deal with the determination of the expectation value and dispersion of the quantity under consideration at asymptotically large times. A one-dimensional case is analyzed in Section 4, where the distribu- tion probability density and the moments of any order are calculated. The concluding Section 5 contains a discussion of the relationship between the volume of the Wiener sausage and the number of different sites visited by a random walk on a lattice. 2. AVERAGE VOLUME OF A WIENER SAUSAGE Let us consider a spherical Brownian particle with a radius b. We introduce (p(r, W,), which is a function of the position r of the point in 1 Karpov Institute of Physical Chemistry, 103064 Moscow, USSR. 333 0022-4715/89/1000-0333506.00/0 Ct" 1989 Plenum Publishing Corporation