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Overview
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 shastic 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 shastic processes and the author's construction of shastic line integrals with Riemannian parallel translation.
Secondly, the book includes applications such as the Langevin equation of statistical mechanics. Nelson's shastic 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 shastic 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, shastic processes, global analysis, analysis on manifolds or differential geometry, and is recommended for graduate level courses.