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Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.
In 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 theory of Aubry-Mather and weak KAM for Hamiltonian systems, we develop an analog inthe setting of linear transfers. We prove the existence of an idempotentoperator which maps into the set of weak KAM solutions, an idempotentlinear transfer that plays the role of the Peierls barrier, and we identifyanalogous objects in this setting such as the Mather measures and the Aubryset. We apply this to the framework of ergodic optimisation in the holonomiccase.
In the first part of this thesis, we study the structure of solutions to the optimal martingale transport problem, when the marginals lie in higher dimensional Euclidean spaces (ℝ^d, d ≥ 2). The problem has been extensively studied in one-dimensional space (ℝ), but few results have been shown in higher dimensions. In this thesis, we propose two conjectures and provide key ideas that lead to solutions in important cases.In the second part, we study the structure of solutions to the optimal subharmonic martingale transport problem, again when the marginals lie in higher dimensional Euclidean spaces. First, we show that this problem has an equivalent formulation in terms of the celebrated Skorokhod embedding problem in probability theory. We then describe the fine structure of the solution provided the marginals are radially symmetric. The general case remains unsolved, and its potential solution calls for a deeper understanding of harmonic analysis and Brownian motion in higher dimensional spaces.
Master's Student Supervision
Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.
In this paper, a self-contained proof is given to a well-known Harnack inequality of second order nondivergent uniformly elliptic operators on Riemannian manifolds with the condition that M-[R(v )]>0, following the ideas of M. Safonov . Basically, the proof consists of three parts: 1)Critical Density Lemma, 2) Power-Decay of the Distribution Functions of Solutions, and 3)Harnack Inequality.
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