| Reference : Geometry and Stochastic Calculus on Wasserstein spaces |
| Dissertations and theses : Doctoral thesis | |||
| Physical, chemical, mathematical & earth Sciences : Mathematics | |||
| http://hdl.handle.net/10993/15561 | |||
| Geometry and Stochastic Calculus on Wasserstein spaces | |
| English | |
| Selinger, Christian [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit] | |
| 8-Nov-2010 | |
| University of Luxembourg, Luxembourg, Luxembourg | |
| Docteur en Mathématiques | |
Thalmaier, Anton  | |
| [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. | |
| http://hdl.handle.net/10993/15561 |
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