Hyperbolic dynamics and Brownian motion. An introduction
type: Item
instance of: scholarly article
zbMATH Open document ID: 1257.37001
author: Jacques Franchi
author: Yves Le Jan
publication date: 2012-08-29T00:00:00Z
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 51M09
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 47D07
zbMATH DE Number: 6073805
zbMATH Keywords: Lorentz-Möbius group \(\text{PSO}(1,d)\)
zbMATH Keywords: Iwasawa and Cartan decompositions
zbMATH Keywords: commutation relations
zbMATH Keywords: Minkowski space
zbMATH Keywords: hyperbolic geometry
zbMATH Keywords: Poincaré coordinates
zbMATH Keywords: stable leaves
zbMATH Keywords: geodesic and horocyclic flows
zbMATH Keywords: Casimir and Laplace operators
zbMATH Keywords: Haar measures
zbMATH Keywords: harmonic measures
zbMATH Keywords: Liouville measures
zbMATH Keywords: Kleinian groups
zbMATH Keywords: ergodicity
zbMATH Keywords: mixing property
zbMATH Keywords: spectral gap
zbMATH Keywords: Brownian motion
zbMATH Keywords: martingales
zbMATH Keywords: Itô's stochastic calculus
zbMATH Keywords: linear stochastic differential equations
zbMATH Keywords: group-valued Brownian motions
zbMATH Keywords: diffusion processes
zbMATH Keywords: hyperbolic and relativistic Brownian motions
zbMATH Keywords: foliated diffusions
zbMATH Keywords: central limit theorem of Sinai
MaRDI profile type: Publication
title: Hyperbolic dynamics and Brownian motion. An introduction
published in: Oxford Mathematical Monographs
review text: The idea of this book is to illustrate the interplay between distinct domains of mathematics, without assuming that the reader is a specialist of any of them.NEWLINENEWLINE Firstly, it provides an elementary introduction to hyperbolic geometry, based on the Lorentz-Möbius group \(\text{PSO}(1,d)\), which plays in relativistic Minkowski's space-time a role analogue to the rotations in Euclidean space; the main difference being the existence of boosts, also called hyperbolic rotations. Presenting hyperbolic geometry via special relativity allows to benefit from the physical intuition. The light cone can be viewed as the set of straight lines which are asymptotic to the unit pseudo-sphere, namely the light rays. The hyperbolic geometry is the geometry of this pseudo-sphere. The boundary of the hyperbolic space is defined as the set of light rays. The key role is thus played by the Lorentz-Möbius group, its Iwasawa decomposition, commutation relations and Haar measure, the hyperbolic Laplacian, and the geodesic and horocyclic flows.NEWLINENEWLINE Secondly, this book introduces basic notions of stochastic analysis: the Wiener process, Itô's stochastic integral and some elements of Itô's calculus. This introduction allows to study linear stochastic differential equations on groups of matrices, and then diffusion processes on homogeneous spaces. In this way are constructed in particular the spherical and hyperbolic Brownian motions, diffusions on the stable leaves, and the Dudley relativistic diffusion (whereas relativity is generally ignored in treaties about stochastic processes).NEWLINENEWLINE Thirdly, quotients of the hyperbolic space under a discrete group of isometries (i.e., in two dimensions, Riemann surfaces endowed with a negative constant curvature metric) are also introduced, and are the framework in which are presented some elements of hyperbolic dynamics, especially the ergodicity of the geodesic and horocyclic flows. This book culminates with an analysis of …
reviewed by: Jacques Franchi
description: scientific article; zbMATH DE number 6073805