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Brownian bridges on Riemannian manifolds
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  • Published: March 1990

Brownian bridges on Riemannian manifolds

  • Pei Hsu1 

Probability Theory and Related Fields volume 84, pages 103–118 (1990)Cite this article

  • 177 Accesses

  • 17 Citations

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Summary

We study properties of Brownian bridges on a complete Riemannian manifoldM. LetQ t x,y be the law of Brownian bridge fromx toy with lifetimet. Q t x,y is a probability measure on the space Ω x,y of continuous paths ω with ω(0)=x and ω(1)=y. We prove thatQ t x,y possesses the large deviation property with the rate function

$$J_{x,y} (\omega ) = \frac{1}{2}\left[ {\int\limits_0^1 {\left| {\dot \omega (s)} \right|^2 ds - \rho (x,y)^2 } } \right].$$

We show that ifM and its metric are analytic then forany x, y onM there exists a probability measure μ x,y which is supported by a subset of the space of minimizing geodesics joiningx andy such that Ω x,y t →μ x,y weakly in Ω x,y ast→0. We also give a complete characterization of the exact support of μ x,y .

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References

  1. Azencott, R.: Grandes déviations et applications. In: Hennequin, P. (ed.) Ecole d'Eté de Probabilités de Saint Flour, VIII. (Lect. Notes Math., vol. 774) Berlin Heidelberg, New York: Springer 1980

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  2. Azencott, R.: Géodesiques et Diffusions en Temps Petit. Astéristique, no. 84–85. Soc. Math. France

  3. Kuo, H-H.: Gaussian measures in Banach spaces. (Lect. Notes Math., vol. 463.) Berlin Heidelberg New York: Springer

  4. Freidlin, M.I., Ventcell, A.D.: Random perturbation of dynamic systems. Berlin, Heidelberg New York: Springer

  5. Molchanov, S.A.: Diffusion processes and Riemannian geometry. Russ. Math. Surv.30, 1–63 (1975)

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Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Illinois at Chicago, 322 Science and Engineering Offices, Box 4348, 60680, Chicago, IL, USA

    Pei Hsu

Authors
  1. Pei Hsu
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Additional information

Research supported in part by the grant NSF-DMS-86-00233. Current address: Department of Math. Northwestern University

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Cite this article

Hsu, P. Brownian bridges on Riemannian manifolds. Probab. Th. Rel. Fields 84, 103–118 (1990). https://doi.org/10.1007/BF01288561

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  • Received: 03 November 1987

  • Revised: 28 April 1989

  • Issue Date: March 1990

  • DOI: https://doi.org/10.1007/BF01288561

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Keywords

  • Manifold
  • Rate Function
  • Stochastic Process
  • Probability Measure
  • Probability Theory
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