Spectral functions of subordinate Brownian motion on closed manifolds

For a class of subordinators we investigate the spectrum of the infinitesimal generator of subordinate Brownian motion on a closed manifold. We consider three spectral functions of the generator: the zeta function, the heat trace and the spectral action. Each spectral function explicitly yields both probabilistic and geometric information, the latter through the classical heat invariants. All constructions are done with classical pseudodifferential operators and are fully analytically tractable.