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