Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1989-0929666-4
MathSciNet review: 929666
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [A] R. Azencott, Diffusions sur les variétés. Généralités, Astérisque 84-85 (1981), 17-32.
  • [BB] L. Bergery and J. Bourguignon, Laplacians and Riemannian submersions with totally geodesic fibres, Illinois J. Math. 26 (1982), 181-200. MR 650387 (84m:58153)
  • [E] K. D. Elworthy, Stochastic differential equations on manifolds, London Math. Soc. Lecture Note Ser., no. 70, Cambridge Univ. Press, 1982. MR 675100 (84d:58080)
  • [EK] K. D. Elworthy and W. S. Kendall, Factorization of harmonic maps and Brownian motions, Local Time to Global Geometry, Control and Physics (K. D. Elworthy, ed.), Pitman Res. Notes in Math. Ser. 150, 1985, pp. 75-83. MR 894524 (88m:58200)
  • [H] R. Hermann, A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle, Proc. Amer. Math. Soc. 11 (1960), 236-242. MR 0112151 (22:3006)
  • [IW] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland, Amsterdam, 1981. MR 1011252 (90m:60069)
  • [K] H. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms, Lecture Notes in Math., vol. 1079, Springer-Verlag, Berlin and New York, pp. 143-303. MR 876080 (87m:60127)
  • [KN] S. Kobayashi and K. Nomizu, Foundations of differential geometry. I, Interscience, New York, 1963. MR 0152974 (27:2945)
  • [O] B. O'Neil, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459-469. MR 0200865 (34:751)
  • [V] J. Vilms, Totally geodesic maps, J. Differential Geometry 4 (1970), 73-79. MR 0262984 (41:7589)
  • [W] B. Watson, Manifold maps commuting with the Laplacian, J. Differential Geometry 8 (1973), 85-94. MR 0365419 (51:1671)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58G32, 53C10

Retrieve articles in all journals with MSC: 58G32, 53C10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0929666-4
Keywords: Diffusions, elliptic operators, fibre bundles, structure groups, connections, Riemannian submersions, totally geodesic fibres
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society