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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Factorization of diffusions on fibre bundles
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by Ming Liao PDF
Trans. Amer. Math. Soc. 311 (1989), 813-827 Request permission

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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Additional 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