  Let $M$ be a differentiable manifold endowed locally with two complementary
distributions, say horizontal and vertical. We consider the two subgroups of
(local) diffeomorphisms of $M$ generated by vector fields in each of of these
distributions. Given a stochastic flow $\varphi_t$ of diffeomorphisms of $M$,
in a neighbourhood of initial condition, up to a stopping time we decompose
$\varphi_t = \xi_t \circ \psi_t$ where the first component is a diffusion in
the group of horizontal diffeomorphisms and the second component is a process
in the group of vertical diffeomorphisms. Further decomposition will include
more than two components; it leads to a maximal cascade decomposition in local
coordinates where each component acts only in the corresponding coordinate.
