Geometry and Stochastic Calculus on Wasserstein spaces

The main object of interest in this thesis is P(M) – the space of probability measures on a manifold endowed with the Wasserstein distance:
In chapter 1 we give the most basic topological facts and introduce a locally convex topology on P∞ (the space of smooth positive densities) to identify this space as infinite dimensional manifold.
In chapter 2 we develop further the Riemannian calculus on P resp. P∞ where the different approaches (calculus of variation, Riemannian geometry on spaces of smooth mappings) are shown to be equivalent on P∞ .
In chapter 3 we restrict ourself tomeasures on the unit circle and give calculations of renormalized Laplacians on the respective Wasserstein spaces, seen as the Hilbert-Schmidt trace of the Hessian: This trace depends on a real parameter s and has an analytic continuation as a function of s ∈ C \ {1} which enables us to calculate evaluate at s = 0: The square-field operator of the Wassersein Laplacian equals the squared Wasserstein gradient times the volume of the unit circle.
In chapter 4 we give an approximation of the Wasserstein space P ([0, 1]) by spaces of box-type measures which are geodesically convex and can be mapped isometrically via a mapping simplex , where a sticky diffusion process is constructed. We show that image of this processes constitute a tight family in C(R_+