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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 provenextending the classical results in Hilbert spaces. Third chapter contains a treatment of the nonsymmetric FokkerPlanck equation as a flow on the Wasserstein2 space of probability measureswe 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 Wasserstein2 spaces over Euclidean spaces is developed. This theory generalizes the theory of gradient flows by AmbrosioGigliSavar. 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.
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