Lorentz-Möbius group \(\text{PSO}(1,d)\)
linear stochastic differential equations
hyperbolic and relativistic Brownian motions
Hyperbolic dynamics and Brownian motion. An introduction
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 the chaotic behaviour of the geodesic flow, which is performed using stochastic analysis methods. This main result is known as Sinai's central limit theorem. Chaotic behaviour arises from the instability of the geodesic flow: small initial perturbations produce a large effect after some time. This follows from the commutation relation between the geodesic flow and either the horocyclic flow or the rotations, which originates in the structure of the Lie algebra of the Lorentz-Möbius group.NEWLINENEWLINE In this book the necessary material from group theory, geometry and stochastic analysis is presented in a self-contained and elementary way. Only basic knowledge of linear algebra, calculus and probability theory is required. General notions of geometry such as manifolds and bundles are avoided, except for the tangent and frame bundles of the hyperbolic space, defined via the embedding in Minkowski's space. The exposition of probabilistic tools is made as short and elementary as possible, to the purpose of making it easily available to analysts and geometers.
scientific article; zbMATH DE number 6073805