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Convexities and optimal transport problems on the Wiener space

2013

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 dime…

Convexité[ MATH.MATH-GM ] Mathematics [math]/General Mathematics [math.GM][MATH.MATH-GM] Mathematics [math]/General Mathematics [math.GM]Monge-Ampère equationConvexityMonge problem[MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM]Dimension infinieTransport optimalLogarithmic concave measureWiener spaceEspace de WienerOptimal transportÉquation de Monge-AmpèreMesure logarithmiquement concaveProblème de MongeInfinite dimension
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