Ordinary and Stochastic Differential Geometry as a Tool for Mathematical Physics

ISBN-13: 9780792341543 / Angielski / Twarda / 1996 / 192 str.

Iu E. Gliklikh; Yuri E. Gliklikh; Y. E. Gliklikh

Opis

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.

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.