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.

Bryant, Sarah. Counting and Partition Function Asymptotics for Subordinate Killed Brownian Motion. In Advances in the Mathematical Sciences: Research from the 2015 Association for Women in Mathematics Symposium, edited by Gail Letzter, Kristin Lauter, Erin Chambers, Nancy Flournoy, Julia Elisenda Grigsby, Carla Martin, Kathleen Ryan, and Konstantina Trivisa, 271-279. Cham: Springer International Publishing: Imprint: Springer, 2016.

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