Heat trace asymptotics of subordinate Brownian motion in Euclidean space

For a class of Laplace exponents we derive the heat trace asymptotics of the generator of the corresponding subordinate Brownian motion on Euclidean space. The terms in the asymptotic expansion are found to depend both on the geometry of Euclidean space and probabilistic properties of the subordinator. The key assumption is the existence of a suitable density for the Levy measure of the subordinator. An intermediate step is the computation of the zeta function of the generator. We employ methods from the theory of classical pseudodifferential operators on Euclidean space. The analysis is highly explicit and fully analytically tractable.