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Ordinary and Stochastic Differential Geometry as a Tool for Mathematical Physics

  • Yuri E. Gliklikh

Part of the Mathematics and Its Applications book series (MAIA, volume 374)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Yuri E. Gliklikh
    Pages 45-74
  3. Yuri E. Gliklikh
    Pages 75-98
  4. Yuri E. Gliklikh
    Pages 99-106
  5. Yuri E. Gliklikh
    Pages 107-136
  6. Yuri E. Gliklikh
    Pages 137-165
  7. Back Matter
    Pages 166-192

About this book

Introduction

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.

Keywords

Probability theory Stochastic processes differential geometry manifold mathematical physics stochastic process

Authors and affiliations

  • Yuri E. Gliklikh
    • 1
  1. 1.Mathematics FacultyVoronezh State UniversityVoronezhRussia

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-015-8634-4
  • Copyright Information Springer Science+Business Media B.V. 1996
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-4731-1
  • Online ISBN 978-94-015-8634-4
  • Buy this book on publisher's site