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We relate Gruet formula for the heat kernel on real hyperbolic spaces to thecommonly used one derived from Millson induction. The bridge between bothformulas is settled by Yor result on the joint distribution of a Brownianmotion and of its exponential functional at fixed time. This result allowsfurther to relate Gruet formula with real parameter to the heat kernel of thehyperbolic Jacobi operator and to derive a new integral representation for theheat kernel of the Maass Laplacian. When applied to harmonic AN groups, Yorresult yields also new a integral representation of their corresponding heatkernels which does not distinguish the parity of the dimension of the center ofthe Lie group N.

Hartman-Watson distribution and hyperbolic-like Heat kernels