Abstract
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 φt of diffeomorphisms of M, in a neighbourhood of an initial condition, up to a stopping time we decompose φt = ξt ◦ ψt 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.
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