Stochastic Equations and Differential Geometry / Edition 1

by Ya.I. Belopolskaya, Yu.L. Dalecky

Overview

'Et moi, ... , si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point aile.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded n- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non­ linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ... '; 'One service logic has rendered com­ puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Product Details

ISBN-13:	9789027728074
Publisher:	Springer Netherlands
Publication date:	02/28/1990
Series:	Mathematics and its Applications , #30
Edition description:	1990
Pages:	260
Product dimensions:	6.10(w) x 9.25(h) x 0.24(d)

Table of Contents

1. Functions and Measures in Linear Spaces.- 1. Spaces, Mappings, Differential Operations.- 2. Measures and Integrals.- 3. Measure Differentiation.- 2. Functions and Measures on Smooth Manifolds.- 1. Smooth Manifolds and Vector Bundles.- 2. Bundle Section, Connections, Differential Operations.- 3. Hilbert and Hilbert-Schmidt Bundles.- 4. Measures on Smooth Manifolds.- 3. Shastic Equations in Banach Spaces.- 1. Basic Notions.- 2. Shastic Integrals for Vector and Operator Functions.- 3. Shastic Equations.- 4. Multiplicative Functionals of Shastic Processes.- 5. Shastic Flow.- 4. Shastic Equations on Smooth Manifolds.- 1. Shastic Differentials.- 2. Shastic Differential Equations on Manifolds.- 3. Shastic Equations in Vector Bundles.- 5. Kolmogorov Equations.- 1. Backward Kolmogorov Equations.- 2. Quasilinear Parabolic Equations.- 3. Forward Kolmogorov Equations.- 6. Diffusion Processes on Lie Groups and Principal Fibre Bundles.- 1. Lie Groups and Smooth Bundles.- 2. Invariant Shastic Equations.- 3. Shastic Equations on Manifolds and their Solution Distribution Properties.- Historical Comments.- References.

Editorial Reviews

Stochastic analysis on manifolds concerns the study, on infinite- dimensional manifolds, of both random processes and partial differential equations, each aspect being covered here. The volume begins with the main tools coming from differential geometry, especially connection theory on bundles. Systematic treatment is given to Ito's stochastic calculus as well as smooth measures, diffusion processes, and invariance properties under smooth group actions. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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