A decomposition of Markov processes via group actions

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We study a decomposition of a general Markov process in a manifold invariant under a Lie group action into a radial part (transversal to orbits) and an angular part (along an orbit). We show that given a radial path, the conditioned angular part is a nonhomogeneous process in a homogeneous space, we obtain a representation of such processes, and as a consequence, we extend the well known skew-product of Euclidean Brownian motion to a general setting.