Originating from lectures by L. Ambrosio at the ETH Zürich in Fall 2001, substantially extended and revised in cooperation with the co-authors, it serves as textbook and reference book on the topic and it is presented as much as possible in a self-contained way. The book contains new results that have never appeared elsewhere and it is devoted to a theory of gradient flows in spaces which are not necessarily endowed with a natural linear or differentiable structure. It consists of two parts, the first one concerning gradient flows in metric spaces and the second one devoted to gradient flows in the space of probability measures on a separable Hilbert space, endowed with the Kantorovich-Rubinstein-Wasserstein distance. The two parts have some connections, due to the fact that the space of probability measures provides an important model to which the "metric" theory applies. However, the book is conceived in such a way that the two parts can be read independently, the first one by the reader more interested in non-smooth analysis and analysis in metric spaces, and the second one by the reader more orientated towards the applications in partial differential equations, measure theory and probability. Written for: Graduate and postgraduate students, researchers
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