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