This book is devoted to a theory of gradient flows in spaces which are not necessarily endowed with a natural linear or differentiable structure. The book originates from lectures by L. Ambrosio at the ETH Zurich in Fall 2001 and contains new results.

Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence Theorems.- Uniqueness, Generation of Contraction Semigroups, Error Estimates.- Gradient Flow in the Space of Probability Measures.- Preliminary Results on Measure Theory.- The Optimal Transportation Problem.- The Wasserstein Distance and its Behaviour along Geodesics.- Absolutely Continuous Curves in p(X) and the Continuity Equation.- Convex Functionals in p(X).- Metric Slope and Subdifferential Calculus in (X).- Gradient Flows and Curves of Maximal Slope in p(X).

Originating from lectures by L. Ambrosio at the ETH Zürich in Fall 2001 Substantially extended and revised in cooperation with the co-authors Serves as textbook and reference book on the topic Presented as much as possible in a self-contained way Containing new results that never appeared elsewhere Includes supplementary material: sn.pub/extras

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