Counting and partition function asymptotics for subordinate killed brownian motion

Abstract

We consider the subordinate killed Brownian motion process generated by first killing Brownian motion at some boundary point on a smooth bounded domain then subordinating by a Lévy time-clock. For classes of subordinators satisfying some growth requirements, we establish asymptotic growth for the eigenvalues associated to these processes. Using an abelian argument we are then able to prove first-term asymptotics for the trace of the heat semigroup, or partition function. For α/ 2 -stable subordinators we prove second-order term asymptotics of the partition function with constants dependent on volume and surface area of the boundary.