Volume nullification and asymptotic flatnessof denerate diffusions

Abstract

LetX t (x) be a singular Process describing a flow of diffeomorphisms ,and kbe a compact surface in with positive finite Hausdorff measure. We present conditions under which the area of Xt:X (k)goes to zero almost surely and in moments as t→∞. It is shown, in particular, that the flowXtX (·) asymptotically nullifies the arc-length of oriented rectifiable arcs . As a consequence, we recover one of our earlier results on the asymptotic flatness of singular diffusions. Though we work with degenerate diffusions, our method applies to the nondegenerate case as well