In Rⁿ, Brownian diffusion leads to the normal or Gaussian distribution. On the sphere Sⁿ, diffusion does not lead to the Fisher distribution which often plays the role of the normal distribution on Sⁿ. On the circle (S¹) and sphere (S²), they are known to be numerically close. It is shown that there exists a random stopping time for the diffusion which leads to the Fisher distribution. This follows from the fact, proved here, that the modified Bessel function Iᵥ(x) is a completely monotone function of v² (for fixed $x > 0$). More generally, we study the class of distributions on Sⁿ which can be represented as mixtures of diffusions. The stopping time distribution is characterized, but not given in computable form. Also, three new distribution functions involving Bessel functions are presented.
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Hartman et al. (1974) studied this question.