[PDF] Conditioned Brownian Motion And Hyperbolic Geodesics In Simply Connected Domains eBook

Author : Jacques Franchi
Publisher : Oxford Mathematical Monographs
Page : 283 pages
File Size : 39,15 MB
Release : 2012-08-16
Category : Mathematics
ISBN : 0199654107

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A simple introduction to several important fields of modern mathematics. The exposition is based on an interplay between hyperbolic geometry, stochastic calculus, special relativity and chaotic dynamics. It is suitable for anyone with some solid background in linear algebra, calculus, and probability theory.

Author : Werner Ballmann
Publisher : Birkhäuser
Page : 114 pages
File Size : 24,97 MB
Release : 2012-12-06
Category : Mathematics
ISBN : 3034892403

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Singular spaces with upper curvature bounds and, in particular, spaces of nonpositive curvature, have been of interest in many fields, including geometric (and combinatorial) group theory, topology, dynamical systems and probability theory. In the first two chapters of the book, a concise introduction into these spaces is given, culminating in the Hadamard-Cartan theorem and the discussion of the ideal boundary at infinity for simply connected complete spaces of nonpositive curvature. In the third chapter, qualitative properties of the geodesic flow on geodesically complete spaces of nonpositive curvature are discussed, as are random walks on groups of isometries of nonpositively curved spaces. The main class of spaces considered should be precisely complementary to symmetric spaces of higher rank and Euclidean buildings of dimension at least two (Rank Rigidity conjecture). In the smooth case, this is known and is the content of the Rank Rigidity theorem. An updated version of the proof of the latter theorem (in the smooth case) is presented in Chapter IV of the book. This chapter contains also a short introduction into the geometry of the unit tangent bundle of a Riemannian manifold and the basic facts about the geodesic flow. In an appendix by Misha Brin, a self-contained and short proof of the ergodicity of the geodesic flow of a compact Riemannian manifold of negative curvature is given. The proof is elementary and should be accessible to the non-specialist. Some of the essential features and problems of the ergodic theory of smooth dynamical systems are discussed, and the appendix can serve as an introduction into this theory.

Author : Cédric Villani
Publisher : Springer Science & Business Media
Page : 970 pages
File Size : 36,48 MB
Release : 2008-10-26
Category : Mathematics
ISBN : 3540710507

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At the close of the 1980s, the independent contributions of Yann Brenier, Mike Cullen and John Mather launched a revolution in the venerable field of optimal transport founded by G. Monge in the 18th century, which has made breathtaking forays into various other domains of mathematics ever since. The author presents a broad overview of this area, supplying complete and self-contained proofs of all the fundamental results of the theory of optimal transport at the appropriate level of generality. Thus, the book encompasses the broad spectrum ranging from basic theory to the most recent research results. PhD students or researchers can read the entire book without any prior knowledge of the field. A comprehensive bibliography with notes that extensively discuss the existing literature underlines the book’s value as a most welcome reference text on this subject.

Author : Christopher J. Bishop
Publisher : Cambridge University Press
Page : 415 pages
File Size : 50,23 MB
Release : 2017
Category : Mathematics
ISBN : 1107134110

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A mathematically rigorous introduction to fractals, emphasizing examples and fundamental ideas while minimizing technicalities.

Author : Steven N. Evans
Publisher : Springer
Page : 205 pages
File Size : 39,68 MB
Release : 2007-09-26
Category : Mathematics
ISBN : 3540747982

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Random trees and tree-valued stochastic processes are of particular importance in many fields. Using the framework of abstract "tree-like" metric spaces and ideas from metric geometry, Evans and his collaborators have recently pioneered an approach to studying the asymptotic behavior of such objects when the number of vertices goes to infinity. This publication surveys the relevant mathematical background and present some selected applications of the theory.