Decomposition of stochastic flows with automorphism of subbundles component

TL;DR

This paper generalizes the decomposition of stochastic flows on manifolds with G-structures, showing that flows can be split into automorphism and transversal components under certain conditions.

Contribution

It introduces a geometric framework for decomposing stochastic flows with respect to G-structures, extending previous Riemannian-based results to more general structures.

Findings

Existence of flow decomposition into automorphism and transversal parts.

Generalization of previous isometric flow decompositions.

Application to manifolds with G-structures beyond Riemannian cases.

Abstract

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 $\phi_t = \xi_t \circ \rho_t$ where $\xi_t \in G(M)$, the remainder $\rho_t$ 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 $\phi_t$ 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 $M$.