Decomposition of stochastic flows in manifolds with complementary distributions

Details Decomposition of stochastic flows in manifolds with complementary distributions Decomposition of stochastic flows in manifolds with complementary distributions

2012-02-24

arxiv.org/abs/1202.5524v1

Mathematics

Dynamical Systems

10 pages

Scientific paper

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.

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