Path and spectral properties of certain Lévy processes
Abstract
This thesis contains several results concerning alpha-stable processes, processes with alpha-stable components, and subordinate killed Brownian motion. We extend a result by Lewis and Li concerning the expected time to see a flat one-dimensional Brownian motion path to include more general processes in higher dimensions. We give exact limits in the case of a d-dimensional process with alpha-stable components for d ≥ 1. We provide the first-order asymptotics of the counting function for subordinate killed Brownian motion for a large class of subordinators. We then give the corresponding partition function asymptotics via the Karamata Tauberian theorem. We also prove second-order asymptotics of the counting function for subordinate killed Brownian motion on domains where the behavior is known for Brownian motion. For an α/2-stable subordinator we prove second-order asymptotics for the partition function. We consider a quantity related to the heat content of alpha-stable processes, proving bounds and limit behavior in terms of the associated spectrum.
Degree
Ph.D.
Advisors
Banuelos, Purdue University.
Subject Area
Mathematics
Off-Campus Purdue Users:
To access this dissertation, please log in to our
proxy server.