{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86944"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86944","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Martingales in Filtering and Geometry","abstract":"We give three applications of martingale theory. First, we study a problem in real-time target tracking. Realistic assumptions, namely limited processing power, turn the variance into a stochastic process. We transform and compensate the variance process so as to obtain a martingale. We find conditions on the parameters under our control that yield a satisfactory tracking mechanism. Specifying the relation between two quantities, we determine an optimal tracking procedure. Second, we give an explicit representation for the solution of the heat equation for trivial vector bundles using Ito's formula and an elementary martingale convergence result. Third, we give a martingale characterization of Yang-Mills fields. This uses stochastic analogues of lasso-forms and integrated lassos. In dimension 4, we relate the Yang-Mills action to the quadratic variation of the martingale used in the characterization. For the special case of self-dual Yang-Mills fields we give an energy identity.","abstract_html":"We give three applications of martingale theory. First, we study a problem in real-time target tracking. Realistic assumptions, namely limited processing power, turn the variance into a stochastic process. We transform and compensate the variance process so as to obtain a martingale. We find conditions on the parameters under our control that yield a satisfactory tracking mechanism. Specifying the relation between two quantities, we determine an optimal tracking procedure. Second, we give an explicit representation for the solution of the heat equation for trivial vector bundles using Ito's formula and an elementary martingale convergence result. Third, we give a martingale characterization of Yang-Mills fields. This uses stochastic analogues of lasso-forms and integrated lassos. In dimension 4, we relate the Yang-Mills action to the quadratic variation of the martingale used in the characterization. For the special case of self-dual Yang-Mills fields we give an energy identity.","abstract_has_math":false,"creators":["Bauer, Robert Otto"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Donald Burkholder"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:20:17Z","date_published":"2015-09-28T15:20:17Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI9737047"],"render_values":[{"text":"(MiAaPQ)AAI9737047","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86944","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Donald Burkholder"]},{"key":"dc:creator","label":"Author","values":["Bauer, Robert Otto"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:20:17Z","10000-01-01","1997"]},{"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/2142/86944","(MiAaPQ)AAI9737047"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We give three applications of martingale theory. First, we study a problem in real-time target tracking. Realistic assumptions, namely limited processing power, turn the variance into a stochastic process. We transform and compensate the variance process so as to obtain a martingale. We find conditions on the parameters under our control that yield a satisfactory tracking mechanism. Specifying the relation between two quantities, we determine an optimal tracking procedure. Second, we give an explicit representation for the solution of the heat equation for trivial vector bundles using Ito's formula and an elementary martingale convergence result. Third, we give a martingale characterization of Yang-Mills fields. This uses stochastic analogues of lasso-forms and integrated lassos. In dimension 4, we relate the Yang-Mills action to the quadratic variation of the martingale used in the characterization. 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First, we study a problem in real-time target tracking. Realistic assumptions, namely limited processing power, turn the variance into a stochastic process. We transform and compensate the variance process so as to obtain a martingale. We find conditions on the parameters under our control that yield a satisfactory tracking mechanism. Specifying the relation between two quantities, we determine an optimal tracking procedure. Second, we give an explicit representation for the solution of the heat equation for trivial vector bundles using Ito's formula and an elementary martingale convergence result. Third, we give a martingale characterization of Yang-Mills fields. This uses stochastic analogues of lasso-forms and integrated lassos. In dimension 4, we relate the Yang-Mills action to the quadratic variation of the martingale used in the characterization. For the special case of self-dual Yang-Mills fields we give an energy identity.","Made available in DSpace on 2015-09-28T15:20:17Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 9737047.pdf: 2281216 bytes, checksum: bb8572559117bf3aa24c5f7b6cdf3eda (MD5) Previous issue date: 1997","Embargo set by: Seth Robbins for item 88225 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","54 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1997."],"dc:identifier":["http://hdl.handle.net/2142/86944","(MiAaPQ)AAI9737047"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["Martingales in Filtering and Geometry"],"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"}