In this paper we study the heat flow on sections of the complex and quaternion line bundles associated to the Hopf fibration over complex and quaternionic projective space. These bundles are equipped with natural metrics and connections. We represent solutions to the heat-equation by the expectation of an appropriate stochastic functional on path space. This uses the stochastic parallel transport. We obtain the first eigenvalue of the horizontal Laplacian by calculating conditional expectations of the stochastic parallel transport. We use a stochastic argument similar to a skew-product decomposition to reduce the problem to an ordinary differential equation of Sturm-Liouville type. We find all eigenvalues and eigenfunctions of this equation.
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