Ordinary and Stochastic Differential Geometry As a Tool for Mathematical Physics - Couverture souple

Gliklikh, Yuri E.

Couverture souple

ISBN 10 : 904814731X ISBN 13 : 9789048147311

Editeur : Springer, 2010

Synopsis

The geometrical methods in modem mathematical physics and the developments in Geometry and Global Analysis motivated by physical problems are being intensively worked out in contemporary mathematics. In particular, during the last decades a new branch of Global Analysis, Stochastic Differential Geometry, was formed to meet the needs of Mathematical Physics. It deals with a lot of various second order differential equations on finite and infinite-dimensional manifolds arising in Physics, and its validity is based on the deep inter-relation between modem Differential Geometry and certain parts of the Theory of Stochastic Processes, discovered not so long ago. The foundation of our topic is presented in the contemporary mathematical literature by a lot of publications devoted to certain parts of the above-mentioned themes and connected with the scope of material of this book. There exist some monographs on Stochastic Differential Equations on Manifolds (e. g. [9,36,38,87]) based on the Stratonovich approach. In [7] there is a detailed description of It6 equations on manifolds in Belopolskaya-Dalecky form. Nelson's book [94] deals with Stochastic Mechanics and mean derivatives on Riemannian Manifolds. The books and survey papers on the Lagrange approach to Hydrodynamics [2,31,73,88], etc., give good presentations of the use of infinite-dimensional ordinary differential geometry in ideal hydrodynamics. We should also refer here to [89,102], to the previous books by the author [53,64], and to many others.

A propos du livre

Présentation de l'éditeur

This book develops new unified methods which lead to results in parts of mathematical physics traditionally considered as being far apart. The emphasis is three-fold:
Firstly, this volume unifies three independently developed approaches to stochastic differential equations on manifolds, namely the theory of Itô equations in the form of Belopolskaya-Dalecky, Nelson's construction of the so-called mean derivatives of stochastic processes and the author's construction of stochastic line integrals with Riemannian parallel translation.
Secondly, the book includes applications such as the Langevin equation of statistical mechanics. Nelson's stochastic mechanics (a version of quantum mechanics), and the hydrodynamics of viscous incompressible fluid treated with the modern Lagrange formalism. Considering these topics together has become possible following the discovery of their common mathematical nature.
Thirdly, the work contains sufficient preliminary and background material from coordinate-free differential geometry and from the theory of stochastic differential equations to make it self-contained and convenient for mathematicians and mathematical physicists not familiar with those branches.
Audience: This volume will be of interest to mathematical physicists, and mathematicians whose work involves probability theory, stochastic processes, global analysis, analysis on manifolds or differential geometry, and is recommended for graduate level courses.

Détails de l'édition

• Éditeur: Springer
• Date d'édition: 2010
• Langue: anglais
• ISBN 10: 904814731X
• ISBN 13: 9789048147311
• Reliure: Broché
• Nombre de pages: 208
• Coordonnées du fabricant: non disponible
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