  We consider mesoscopic thermal transport between two bulk dielectrics joined
by a narrow wire or weak mechanical link. In the ``tunneling'' regime where the
phonon transmission probability through the link is small and the thermal
conductance is much less than the value for a single channel, the thermal
current is determined by a product of the local vibrational spectral densities
of the two bodies. We derive an expression for the thermal current that is a
thermal analog of the well-known formula for the electrical current through a
tunneling barrier.
