Factorization of Diffusions on Fibre Bundles

Abstract

Let $\pi :M \to N$ be a fibre bundle with a $G$-structure and a connection. A $G$-invariant operator $A$ on the standard fibre $F$ is "shifted" to an operator ${A^{\ast }}$ on $M$ and a semielliptic operator $B$ on $N$ is "lifted" to an operator $\tilde B$ on $M$. Let ${X_t}$ be an $A$-diffusion on $F$, let ${Y_t}$ be a $B$-diffusion on $N$ which is independent of ${X_t}$ and let ${\Psi _t}$ be its horizontal lift in the associated principal bundle. Then ${Z_t} = {\Psi _t}({X_t})$ is a diffusion on $M$ with generator ${A^{\ast }} + \tilde B$. Conversely, such a factorization is possible only if the fibre bundle has a proper $G$-structure. In the case of a Riemannian submersion, $X,\;Y$ and $Z$ can be taken to be Brownian motions and the existence of a $G$-structure then means that the fibres are totally geodesic.