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The heat kernel formula in a geodesic chart and some applications to the eigenvalue problem of the 3-sphere
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  • Published: September 1991

The heat kernel formula in a geodesic chart and some applications to the eigenvalue problem of the 3-sphere

  • Martin Ngu Ndumu1 

Probability Theory and Related Fields volume 88, pages 343–361 (1991)Cite this article

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Summary

This paper deals with a heat kernel formula in a geodesic chart with some applications to the standardn-sphere. Our emphasis will be on the special case of the 3-sphere which exhibits some identities linking spherical harmonics and certain homogeneous polynomials harmonic on ℝ4. In particular, we will deduce an expression forP x (ζ>t) where ζ is the first (random) time that the bridge process inS 3 hits the south pole. Another easy consequence will be a special case of the H.P. McKean and I.M. Singer expansion of the heat kernel.

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Authors and Affiliations

  1. Department of Mathematics, Yaoundé University, B.P.812, Yaounde, Cameroon

    Martin Ngu Ndumu

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  1. Martin Ngu Ndumu
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Additional information

This work is part of a Ph.D. Thesis undertaken under Professor K.D. Elworthy, Mathematics Institute, Warwick University, Coventry CV4 7AL, England

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Cite this article

Ndumu, M.N. The heat kernel formula in a geodesic chart and some applications to the eigenvalue problem of the 3-sphere. Probab. Th. Rel. Fields 88, 343–361 (1991). https://doi.org/10.1007/BF01418865

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  • Received: 17 April 1989

  • Revised: 09 March 1990

  • Issue Date: September 1991

  • DOI: https://doi.org/10.1007/BF01418865

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Keywords

  • Stochastic Process
  • Probability Theory
  • Eigenvalue Problem
  • Mathematical Biology
  • Spherical Harmonic
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