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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

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