Decomposition of stochastic flows along complementary distributions

Given two complementary distributions in the tangent bundle of a manifold, we find conditions to factorize an stochastic flow into a diffusion in the (infinite dimensional) Lie group of diffeomorphisms which preserve one distribution (horizontal), composed with a process in the Lie group of diffemorphisms which preserve the other distribution (vertical). This decomposition generalizes previous approach, e.g. using coordinate maps, by Ming Liao and others.