{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86854"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86854","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Potential Theory of Generalized Hyperbolic Processes","abstract":"Let Yt be a rotationally invariant generalized hyperbolic process in Rd , d ≥ 3. Yt can be obtained by subordinating Brownian motion with a generalized inverse Gaussian subordinator Tt. We introduce generalized inverse Gaussian and generalized hyperbolic processes in chapter 1. In chapter 2, we study the asymptotic behaviors of the Green function of Y t near zero and infinite, and jumping function of Yt near zero. We prove that Harnack inequality is valid for nonnegative harmonic functions of Yt. In chapter 3, we show that the rotationally invariant generalized hyperbolic processes in a bounded C1,1 open set D can be obtained from rotationally invariant Cauchy processes in D through a combination of a pure jump Girsanov transform and a Feynman-Kac transform. From this, we deduce that the Green functions for these two processes in D are comparable, and the sharp estimate of the Green function of Yt in D is given. In the last chapter, we show boundary Harnack principle holds true for rotationally invariant generalized hyperbolic processes.","abstract_html":"Let Yt be a rotationally invariant generalized hyperbolic process in Rd , d ≥ 3. Yt can be obtained by subordinating Brownian motion with a generalized inverse Gaussian subordinator Tt. We introduce generalized inverse Gaussian and generalized hyperbolic processes in chapter 1. In chapter 2, we study the asymptotic behaviors of the Green function of Y t near zero and infinite, and jumping function of Yt near zero. We prove that Harnack inequality is valid for nonnegative harmonic functions of Yt. In chapter 3, we show that the rotationally invariant generalized hyperbolic processes in a bounded C1,1 open set D can be obtained from rotationally invariant Cauchy processes in D through a combination of a pure jump Girsanov transform and a Feynman-Kac transform. From this, we deduce that the Green functions for these two processes in D are comparable, and the sharp estimate of the Green function of Yt in D is given. In the last chapter, we show boundary Harnack principle holds true for rotationally invariant generalized hyperbolic processes.","abstract_has_math":false,"creators":["Wang, Yun"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Song, Renming"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:19:52Z","date_published":"2015-09-28T15:19:52Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3199168"],"render_values":[{"text":"(MiAaPQ)AAI3199168","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86854","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Song, Renming"]},{"key":"dc:creator","label":"Author","values":["Wang, Yun"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:19:52Z","10000-01-01","2005"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86854","(MiAaPQ)AAI3199168"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let Yt be a rotationally invariant generalized hyperbolic process in Rd , d ≥ 3. Yt can be obtained by subordinating Brownian motion with a generalized inverse Gaussian subordinator Tt. We introduce generalized inverse Gaussian and generalized hyperbolic processes in chapter 1. In chapter 2, we study the asymptotic behaviors of the Green function of Y t near zero and infinite, and jumping function of Yt near zero. We prove that Harnack inequality is valid for nonnegative harmonic functions of Yt. In chapter 3, we show that the rotationally invariant generalized hyperbolic processes in a bounded C1,1 open set D can be obtained from rotationally invariant Cauchy processes in D through a combination of a pure jump Girsanov transform and a Feynman-Kac transform. From this, we deduce that the Green functions for these two processes in D are comparable, and the sharp estimate of the Green function of Yt in D is given. In the last chapter, we show boundary Harnack principle holds true for rotationally invariant generalized hyperbolic processes.","Made available in DSpace on 2015-09-28T15:19:52Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3199168.pdf: 2399420 bytes, checksum: 5015283d476c287ce5833fe13c5b88dc (MD5) Previous issue date: 2005","Embargo set by: Seth Robbins for item 88135 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","87 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2005."]},{"key":"dc:title","label":"Title","values":["Potential Theory of Generalized Hyperbolic Processes"]}]}],"canonical_facts":{"dc:contributor":["Song, Renming"],"dc:creator":["Wang, Yun"],"dc:date":["2015-09-28T15:19:52Z","10000-01-01","2005"],"dc:description":["Let Yt be a rotationally invariant generalized hyperbolic process in Rd , d ≥ 3. Yt can be obtained by subordinating Brownian motion with a generalized inverse Gaussian subordinator Tt. We introduce generalized inverse Gaussian and generalized hyperbolic processes in chapter 1. In chapter 2, we study the asymptotic behaviors of the Green function of Y t near zero and infinite, and jumping function of Yt near zero. We prove that Harnack inequality is valid for nonnegative harmonic functions of Yt. In chapter 3, we show that the rotationally invariant generalized hyperbolic processes in a bounded C1,1 open set D can be obtained from rotationally invariant Cauchy processes in D through a combination of a pure jump Girsanov transform and a Feynman-Kac transform. From this, we deduce that the Green functions for these two processes in D are comparable, and the sharp estimate of the Green function of Yt in D is given. In the last chapter, we show boundary Harnack principle holds true for rotationally invariant generalized hyperbolic processes.","Made available in DSpace on 2015-09-28T15:19:52Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3199168.pdf: 2399420 bytes, checksum: 5015283d476c287ce5833fe13c5b88dc (MD5) Previous issue date: 2005","Embargo set by: Seth Robbins for item 88135 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","87 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2005."],"dc:identifier":["http://hdl.handle.net/2142/86854","(MiAaPQ)AAI3199168"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["Potential Theory of Generalized Hyperbolic Processes"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:28Z"}