Translation and dispersion of mass by isotropic Brownian flows

We study the long-term translation and dispersion of a mass distribution carried by an isotropic Brownian flow on Rd. We use the variance of the center of mass as a measure of translation and the mean of the centered spatial second moments as a measure of dispersion. We find the exact growth rates of these quantities in essentially all cases. When the mass distribution is diffuse, the rates of growth depend strongly on the degree of compressibility in the flow. For example, in two dimensions we show that the variance of the center of mass grows more slowly than logarithmically in the incompressible case, while it grows linearly in highly compressible flows. Our method involves a detailed analysis of the two-point separation process and uses methods developed in a companion paper for the analysis of mean occupation times of one-dimensional Markov processes.