Ordinary and Stochastic Differential Geometry as a Tool for Mathematical Physics

ISBN-13: 9789048147311 / Angielski / Miękka / 2010 / 192 str.

Yuri E. Gliklikh

Opis

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 Ito 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.

Recenzje

` ... the author has successfully arranged them [the topics] into a smooth, coherent rexposition. It is highly recommended for those readers with good background knowledge in differential geometry, stochastic analysis, and mathematical physics.'
Mathematical Reviews, 98k

Spis treści

Introduction. I. Elements of Coordinate-Free Differential Geometry. II. Introduction to Stochastic Analysis in Rn III. Stochastic Differential Equations on Manifolds. IV. Langevin's Equation in Geometric Form. V. Nelson's Stochastic Mechanics. VI. The Lagrangian Approach to Hydrodynamics. Appendix: Solution of the Newton-Nelson Equation with Random Initial Data; Yu.E. Glikhlikh, T.J. Zastawniak. References. Index.