Reference : Geometry and Stochastic Calculus on Wasserstein spaces
Dissertations and theses : Doctoral thesis
Physical, chemical, mathematical & earth Sciences : Mathematics
Geometry and Stochastic Calculus on Wasserstein spaces
Selinger, Christian [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit]
University of Luxembourg, ​Luxembourg, ​​Luxembourg
Docteur en Mathématiques
Thalmaier, Anton mailto
[en] Optimal transport Histograms ; Regularized Laplacian Simplex ; Wasserstein space Infinite dimensional diffusion process 8
[en] 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_+
[en] P ([0, 1])) with respect to the Skorohod topology.
In the last chapter we restrict ourselves to the space of histograms on the unit interval. We calculate the Wasserstein distances numerically and obtain a Riemannian metric on the simplex. We investigate explosion behaviour of the respective diffusion processes in dimension 1 and 2.

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