Geometric approach to evolution problems in metric spaces - Stojkovic, Igor
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This PhD thesis contains four chapters where research material on a range of different topics is presented. The used and developed techniques fall within the scope of analysis, probability and metric geometry, while a significant part of the manuscript contributes to the optimal transportation theory. In the second chapter the product formulas for semigroups induced by convex functionals in general CAT(0) spaces are proven---extending the classical results in Hilbert spaces. Third chapter contains a treatment of the non-symmetric Fokker-Planck equation as a flow on the Wasserstein-2 space of…mehr

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Produktbeschreibung
This PhD thesis contains four chapters where research material on a range of different topics is presented. The used and developed techniques fall within the scope of analysis, probability and metric geometry, while a significant part of the manuscript contributes to the optimal transportation theory. In the second chapter the product formulas for semigroups induced by convex functionals in general CAT(0) spaces are proven---extending the classical results in Hilbert spaces. Third chapter contains a treatment of the non-symmetric Fokker-Planck equation as a flow on the Wasserstein-2 space of probability measures---we prove that its semigroup of solutions possesses similar properties to the properties of the gradient flow semigroups. In the forth chapter a general theory of maximal monotone operators and the induced flows on Wasserstein-2 spaces over Euclidean spaces is developed. This theory generalizes the theory of gradient flows by Ambrosio-Gigli-Savaré. In the fifth chapter the existence of an invariant measure for stochastic delay equations is proven. The diffusion coefficient has an exponentially stable delay, and is only assumed to be locally Lipschitz and bounded.
Autorenporträt
Igor Stojkovi¿ was born in 1972, in Belgrade. Shortly after completing his high school education and due to the outbreak of the civil war he fled to The Netherlands. After receiving his residence permit eight years later he resumed his studies of Mathematics. He has defended his PhD dissertation successfully in April 2011 at the Leiden University.