Mathematical Physics

Download E-books Hyperbolic Dynamics and Brownian Motion: An Introduction (Oxford Mathematical Monographs) PDF

By Jacques Franchi

Hyperbolic Dynamics and Brownian movement illustrates the interaction among specified domain names of arithmetic. there isn't any assumption that the reader is a expert in any of those domain names: merely easy wisdom of linear algebra, calculus and chance concept is needed.

The content material might be summarized in 3 ways:

Firstly, this e-book offers an advent to hyperbolic geometry, in keeping with the Lorentz crew. The Lorentz workforce performs, in relativistic space-time, a job analogue to the rotations in Euclidean area. The hyperbolic geometry is the geometry of the unit pseudo-sphere. The boundary of the hyperbolic house is outlined because the set of sunshine rays. particular consciousness is given to the geodesic and horocyclic flows. Hyperbolic geometry is gifted through exact relativity to learn from the actual intuition.

Secondly, this publication introduces uncomplicated notions of stochastic research: the Wiener method, Ito's stochastic vital, and calculus. This advent permits research in linear stochastic differential equations on teams of matrices. during this means the round and hyperbolic Brownian motions, diffusions at the reliable leaves, and the relativistic diffusion are developed.

Thirdly, quotients of the hyperbolic area less than a discrete team of isometries are brought. during this framework a few components of hyperbolic dynamics are provided, because the ergodicity of the geodesic and horocyclic flows. This ebook culminates with an research of the chaotic behaviour of the geodesic movement, played utilizing stochastic research tools. This major result's often called Sinai's relevant restrict theorem.

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