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Convexities and optimal transport problems on the Wiener space

Abstract : The aim of this PhD is to study the optimal transportation theory in some abstract Wiener space. You can find the results in four main parts and they are aboutThe convexity of the relative entropy. We will extend the well known results in finite dimension to the Wiener space, endowed with the uniform norm. To be precise the relative entropy is (at least weakly) geodesically 1-convex in the sense of the optimal transportation in the Wiener space.The measures with logarithmic concave density. The first important result consists in showing that the Harnack inequality holds for the semi-group induced by such a measure in the Wiener space. The second one provides us a finite dimensional and dimension-free inequality which gives estimate on the difference between two optimal maps.The Monge Problem. We will be interested in the Monge Problem on the Wiener endowed with different norms: either some finite valued norms or the pseudo-norm of Cameron-Martin.The Monge-Ampère equation. Thanks to the inequalities obtained above, we will be able to build strong solutions of the Monge-Ampère (those which are induced by the quadratic cost) equation on the Wiener space, provided the considered measures satisfy weak conditions
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Submitted on : Thursday, January 16, 2014 - 11:57:08 AM
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Vincent Nolot. Convexities and optimal transport problems on the Wiener space. General Mathematics [math.GM]. Université de Bourgogne, 2013. English. ⟨NNT : 2013DIJOS016⟩. ⟨tel-00932092⟩



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