## Factorization of diffusions on fibre bundles

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- by Ming Liao
- Trans. Amer. Math. Soc.
**311**(1989), 813-827 - DOI: https://doi.org/10.1090/S0002-9947-1989-0929666-4
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## 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.## References

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## Bibliographic Information

- © Copyright 1989 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**311**(1989), 813-827 - MSC: Primary 58G32; Secondary 53C10
- DOI: https://doi.org/10.1090/S0002-9947-1989-0929666-4
- MathSciNet review: 929666