The aim of this article is to discuss a volume nullification property of thediffusion process determined by a stochastic differential equation on a manifold. LetX_t(x) be a diffusion process describing a flow of diffeomorphisms x→X_t (x) in a manifoldM, and K be a compact surface in M with positive finite Hausdorff measure. We presentconditions under which the area of X_t(K) goes to zero almost surely and in momentsas t→∞! in particular, the flow X_t(·) asymptotic nullifies the arc-lenth of orientedrectifiable arcs r: [0, 1]→M.
This paper introdnces some concepts of conditional stability of stochasticVolterra equations with anticipating kernel. Snfficient conditions of these types of sta-bility are established via Lyapunov funciton.
