Methods of global analysis and stochastic analysis are most often applied in mathematical physics as separate entities, thus forming important directions in the field. However, while combination of the two subject areas is rare, it is fundamental for the consideration of a broader class of problems.
This book develops methods of Global Analysis and Stochastic Analysis such that their combination allows one to have a more or less common treatment for areas of mathematical physics that traditionally are considered as divergent and requiring different methods of investigation.
Global and Stochastic Analysis with Applications to Mathematical Physics covers branches of mathematics that are currently absent in monograph form. Through the demonstration of new topics of investigation and results, both in traditional and more recent problems, this book offers a fresh perspective on ordinary and stochastic differential equations and inclusions (in particular, given in terms of Nelson's mean derivatives) on linear spaces and manifolds. Topics covered include classical mechanics on non-linear configuration spaces, problems of statistical and quantum physics, and hydrodynamics.
A self-contained book that provides a large amount of preliminary material and recent results which will serve to be a useful introduction to the subject and a valuable resource for further research. It will appeal to researchers, graduate and PhD students working in global analysis, stochastic analysis and mathematical physics.
Table of Contents
Part I Global Analysis.- Manifolds and Related Objects.- Connections.- Ordinary Differential Equations.- Elements of the Theory of Set-Valued Mappings.- Analysis on Groups of Diffeomorphisms.- Part II Stochastic Analysis.- Essentials from Stochastic Analysis in Linear Spaces.- Stochastic Analysis on Manifolds.- Mean Derivatives in Linear Spaces.- Mean Derivatives on Manifolds.- Stochastic Analysis on Groups of Diffeomorphism.- Part III Applications to Mathematical Physics.- Newtonian Mechanics.- Accessible Points and Sub-Manifolds of Mechanical Systems. Controllability.- Some Problems on Lorentz Manifolds.- Mechanical Systems with Random Perturbations.- The Newton-Nelson Equation.- Hydrodynamics.