  We consider global geometric properties of a codimension one manifold
embedded in Euclidean space, as it evolves under an isotropic and volume
preserving Brownian flow of diffeomorphisms. In particular, we obtain
expressions describing the expected rate of growth of the Lipschitz-Killing
curvatures, or intrinsic volumes, of the manifold under the flow.
  These results shed new light on some of the intriguing growth properties of
flows from a global perspective, rather than the local perspective, on which
there is a much larger literature.
