Decomposition of stochastic flows with automorphism of subbundles component

We show that given a $G$-structure $P$ on a differentiable manifold $M$, if the group $G(M)$ of automorphisms of $P$ is big enough, then there exists the quotient of an stochastic flows $phi_t$ by $G(M)$, in the sense that $ϕ_t = ξ_t \circ ρ_t$ where $ξ_t \in G(M)$, the remainder $ρ_t$ has derivative which is vertical but transversal to the fibre of $P$. This geometrical context generalizes previous results where $M$ is a Riemannian manifold and $ϕ_t$ is decomposed with an isometric component, see Liao \cite{Liao1} and Ruffino \cite{Ruffino}, which in our context corresponds to the particular case of an SO(n)-structure on $M$.

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