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Books > Science & Mathematics > Physics > Relativity physics
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Hyperbolic Dynamics and Brownian Motion - An Introduction (Hardcover)
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Hyperbolic Dynamics and Brownian Motion - An Introduction (Hardcover)
Series: Oxford Mathematical Monographs
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Hyperbolic Dynamics and Brownian Motion illustrates the interplay
between distinct domains of mathematics. There is no assumption
that the reader is a specialist in any of these domains: only basic
knowledge of linear algebra, calculus and probability theory is
required. The content can be summarized in three ways: Firstly,
this book provides an introduction to hyperbolic geometry, based on
the Lorentz group. The Lorentz group plays, in relativistic
space-time, a role analogue to the rotations in Euclidean space.
The hyperbolic geometry is the geometry of the unit pseudo-sphere.
The boundary of the hyperbolic space is defined as the set of light
rays. Special attention is given to the geodesic and horocyclic
flows. Hyperbolic geometry is presented via special relativity to
benefit from the physical intuition. Secondly, this book introduces
basic notions of stochastic analysis: the Wiener process, Ito's
stochastic integral, and calculus. This introduction allows study
in linear stochastic differential equations on groups of matrices.
In this way the spherical and hyperbolic Brownian motions,
diffusions on the stable leaves, and the relativistic diffusion are
constructed. Thirdly, quotients of the hyperbolic space under a
discrete group of isometries are introduced. In this framework some
elements of hyperbolic dynamics are presented, as the ergodicity of
the geodesic and horocyclic flows. This book culminates with an
analysis of the chaotic behaviour of the geodesic flow, performed
using stochastic analysis methods. This main result is known as
Sinai's central limit theorem.
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