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Semimartingales and geometric inequalities on locally symmetric manifolds
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  • Published: 13 October 2007

Semimartingales and geometric inequalities on locally symmetric manifolds

  • H. Le1 &
  • D. Barden2 

Probability Theory and Related Fields volume 142, pages 285–311 (2008)Cite this article

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  • 2 Citations

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Abstract

We generalise, to complete, connected and locally symmetric Riemannian manifolds, the construction of coupled semimartingales X and Y given in Le and Barden (J Lond Math Soc 75:522–544, 2007). When such a manifold has non-negative curvature, this makes it possible for the stochastic anti-development of the corresponding semimartingale \({\rm exp}_{X_t} \big(\alpha\,{\rm exp}^{-1}_{X_t}(Y_t)\big)\) to be a time-changed Brownian motion with drift when X and Y are. As an application, we use the latter result to strengthen, and extend to locally symmetric spaces, the results of Le and Barden (J Lond Math Soc 75:522–544, 2007) concerning an inequality involving the solutions of the parabolic equation \(\frac{\partial\psi} {\partial t} = \frac{1}{2}\Delta\psi - h\,\psi\) with Dirichlet boundary condition and an inequality involving the first eigenvalues of the Laplacian, both on three related convex sets.

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References

  1. Borell C. (2000). Diffusion equations and geometric inequalities. Potential Anal. 12: 49–71

    Article  MATH  MathSciNet  Google Scholar 

  2. Cordero-Erausquin D., McCann R.J. and Schmuckenschläger M. (2001). A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146: 219–257

    Article  MATH  MathSciNet  Google Scholar 

  3. Cranston M. (1991). Gradient estimates on manifolds using coupling. J. Funct. Anal. 99: 110–124

    Article  MATH  MathSciNet  Google Scholar 

  4. Gallot S., Hulin D. and Lafontaine J. (1987). Riemannian Geometry. Springer, Heidelberg

    MATH  Google Scholar 

  5. Jost J. (1998). Riemannian Geometry and Geometric Analysis. Springer, Heidelberg

    MATH  Google Scholar 

  6. Kendall W.S. (1986). Stochastic differential geometry, a coupling property and harmonic maps. J. Lond. Math. Soc. 33: 554–566

    Article  MATH  MathSciNet  Google Scholar 

  7. Kendall W.S. (1987). Stochastic differential geometry: an introduction. Acta Appl. Math. 9: 29–60

    Article  MATH  MathSciNet  Google Scholar 

  8. Kendall, W.S.: Martingales on manifolds and harmonic maps. In: Durrett, R., Pinsky, M. (eds.) The Geometry of Random Motion, pp. 121–157. American Mathematical Society, Providence (1988)

    Google Scholar 

  9. Kobayshi S. and Nomizu K. (1969). Foundations of Differential Geometry, vol. 2. Wiley, London

    Google Scholar 

  10. Le H. and Barden D. (2007). Semimartingales and geometric inequalities on manifolds. J. Lond. Math. Soc. 75: 522–544

    Article  MATH  MathSciNet  Google Scholar 

  11. O’Neill B. (1983). Semi-Riemannian Geometry. Academic, London

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Science, University of Nottingham, University Park, NG7 2RD, Nottingham, UK

    H. Le

  2. University of Cambridge, Cambridge, UK

    D. Barden

Authors
  1. H. Le
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  2. D. Barden
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Correspondence to H. Le.

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Cite this article

Le, H., Barden, D. Semimartingales and geometric inequalities on locally symmetric manifolds. Probab. Theory Relat. Fields 142, 285–311 (2008). https://doi.org/10.1007/s00440-007-0105-y

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  • Received: 14 December 2006

  • Revised: 10 September 2007

  • Published: 13 October 2007

  • Issue Date: September 2008

  • DOI: https://doi.org/10.1007/s00440-007-0105-y

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Keywords

  • Brownian motion
  • First eigenvalue of the Laplacian
  • Jacobi field
  • Parabolic equation
  • Parallel translation

Mathematics Subject Classification (2000)

  • 58J65
  • 58J32
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