Heat trace asymptotics of subordinate Brownian motion in Euclidean space

TL;DR

This paper derives explicit heat trace asymptotics for subordinate Brownian motions in Euclidean space, linking geometric and probabilistic properties through pseudodifferential operator methods.

Contribution

It provides a novel explicit analysis of heat trace asymptotics for a class of subordinate Brownian motions using pseudodifferential operator techniques.

Findings

Asymptotic expansion terms depend on Euclidean geometry and subordinator properties.

Computed the zeta function of the generator explicitly.

Established conditions on Levy measure density for analysis.

Abstract

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.