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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Factorization of diffusions on fibre bundles

Author: Ming Liao
Journal: Trans. Amer. Math. Soc. 311 (1989), 813-827
MSC: Primary 58G32; Secondary 53C10
MathSciNet review: 929666
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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.

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Keywords: Diffusions, elliptic operators, fibre bundles, structure groups, connections, Riemannian submersions, totally geodesic fibres
Article copyright: © Copyright 1989 American Mathematical Society