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Second order analysis on $(\mathscr P_2(M),W_2)$
About this Title
Nicola Gigli, University of Bordeaux
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 216, Number 1018
ISBNs: 978-0-8218-5309-2 (print); 978-0-8218-8529-1 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00619-2
Published electronically: June 21, 2011
Keywords: Wesserstein distance,
weak Riemannian structure
MSC: Primary 53C15, 49Q20
Table of Contents
Chapters
- Introduction
- 1. Preliminaries and notation
- 2. Regular curves
- 3. Absolutely continuous vector fields
- 4. Parallel transport
- 5. Covariant derivative
- 6. Curvature
- 7. Differentiability of the exponential map
- 8. Jacobi fields
- A. Density of regular curves
- B. $C^1$ curves
- C. On the definition of exponential map
- D. A weak notion of absolute continuity of vector fields
Abstract
We develop a rigorous second order analysis on the space of probability measures on a Riemannian manifold endowed with the quadratic optimal transport distance $W_2$. Our discussion comprehends: definition of covariant derivative, discussion of the problem of existence of parallel transport, calculus of the Riemannian curvature tensor, differentiability of the exponential map and existence of Jacobi fields. This approach does not require any smoothness assumption on the measures considered.- Luigi Ambrosio and Nicola Gigli, Construction of the parallel transport in the Wasserstein space, Methods Appl. Anal. 15 (2008), no. 1, 1–29. MR 2482206, DOI 10.4310/MAA.2008.v15.n1.a3
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