Linear transfers, Kantorovich operators, and their ergodic properties
department or school: Science, Faculty of
department or school: Mathematics, Department of
data provider: DSpace
degree campus: UBCV
creator: Bowles, Malcolm
issued: 2020-08-25T14:41:30Z
issued: 2020
related degree: Doctor of Philosophy - PhD
degree grantor: University of British Columbia
description: In this work, we introduce and study a class of convex functionals on pairs of probability measures, the linear transfers, which have a structure that com- monly arises in the dual formulations of many well-studied variational prob- lems. We show that examples of linear transfers include a large number of well-known transport problems, including the weak, stochastic, martingale, and cost-minimising transports. Further examples include the balayage of measures, and ergodic optimisation of expanding dynamical systems, among others. We also introduce an extension of the linear transfers, the convex transfers, and show that they include the relative entropy functional and p-powers (p ≥ 1) of linear transfers. We study the properties of linear and convex transfers and show that the inf-convolution operation preserves their structure. This allows dual formu- lations of transport-entropy and other related inequalities, to be computed in a systematic fashion. Motivated by connections of optimal transport to the theory of Aubry- Mather and weak KAM for Hamiltonian systems, we develop an analog in the setting of linear transfers. We prove the existence of an idempotent operator which maps into the set of weak KAM solutions, an idempotent linear transfer that plays the role of the Peierls barrier, and we identify analogous objects in this setting such as the Mather measures and the Aubry set. We apply this to the framework of ergodic optimisation in the holonomic case.
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/75672
note: Linear transfers, Kantorovich operators, and theirergodic propertiesbyMalcolm BowlesB. Sc. University of Victoria, 2012M. Sc, University of Victoria, 2014a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoralstudies(Mathematics)The University of British Columbia(Vancouver)August 2020c© Malcolm Bowles, 2020The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Linear transfers, Kantorovich operators, and their ergodicpropertiessubmitted by Malcolm Bowles in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Mathematics.Examining Committee:Nassif Ghoussoub, MathematicsSupervisorYoung-Heon Kim, MathematicsSupervisorPhilip Loewen, MathematicsSupervisory Committee MemberStephen Gustafson, MathematicsUniversity ExaminerGordon Semenoff, PhysicsUniversity ExamineriiAbstractIn this work, we introduce and study a class of convex functionals on pairs ofprobability measures, the linear transfers, which have a structure that com-monly arises in the dual formulations of many well-studied variational prob-lems. We show that examples of linear transfers include a large number ofwell-known transport problems, including the weak, stochastic, martingale,and cost-minimising transports. Further examples include the balayage ofmeasures, and ergodic optimisation of expanding dynamical systems, amongothers. We also introduce an extension of the linear transfers, the convextransfers, and show that they include the relative entropy functional andp-powers (p ≥ 1) of linear transfers.We study the properties of linear and convex transfers and show that theinf-convolution operation preserves their structure. This allows dual formu-lations of transport-entropy and other related inequalities, to be computedin a systematic fashion.Motivated by connections of optimal transport to the the…
has type: Thesis/Dissertation
date issued: 2020-11
is shown at: 10.14288/1.0392952
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
type: Text
identifier URI: http://hdl.handle.net/2429/75672