Stochastic Equations and Differential Geometry

ISBN-13: 9789401074933 / Angielski / Miękka / 2014 / 260 str.

Ya.I. Belopolskaya;Yu.L. Dalecky

Opis

'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.

Spis treści

1. Functions and Measures in Linear Spaces.- 1. Spaces, Mappings, Differential Operations.- 1.1. Spaces and Operators.- 1.2. Differentiable Functions.- 1.3. Differentiable Equations.- 2. Measures and Integrals.- 2.1. Measure Spaces, Integrals.- 2.2. Measures in Linear Spaces.- 3. Measure Differentiation.- 3.1. Logarithmic Derivatives.- 3.2. Estimates of Logarithmic Derivative Powers.- 3.3. Higher-Order Differential Operations.- 3.4. Smooth Measure Mappings.- 2. Functions and Measures on Smooth Manifolds.- 1. Smooth Manifolds and Vector Bundles.- 1.1. Banach Manifolds.- 1.2. Vector Bundles.- 1.3. Tangent Bundle.- 2. Bundle Section, Connections, Differential Operations.- 2.1. Bundle Sections.- 2.2. Connection Mapping.- 2.3. Parallel Displacement.- 2.4. Induced Connections.- 2.5. Covariant Differentiation of Sections.- 2.6. Tangent Bundle Connection and Exponential Mapping.- 3. Hilbert and Hilbert-Schmidt Bundles.- 4. Measures on Smooth Manifolds.- 4.1. Logarithmic Derivative.- 4.2. Higher-Order Differential Operations, Measure Mapping.- 3. Stochastic Equations in Banach Spaces.- 1. Basic Notions.- 1.1. Random Variables. Independence.- 1.2. Moments, Characteristic Functions, Conditional Expectation.- 1.3. Random Functions, Markov Processes.- 1.4. Real Wiener Processes and Stochastic Wiener Integrals.- 2. Stochastic Integrals for Vector and Operator Functions.- 2.1. Vector Wiener Process.- 2.2. Random Function Stochastic Integral.- 2.3. Estimates of Banach Space-Valued Gaussian Random Variables.- 2.4. Stochastic Integral Properties. Stochastic Differentials.- 3. Stochastic Equations.- 3.1. Existence and Uniqueness Theorem for Solutions of Stochastic Equations in Banach Spaces.- 4. Multiplicative Functionals of Stochastic Processes.- 4.1. The Simplest Situation. Main Definitions.- 4.2. Linear Stochastic Equations.- 4.3. Stochastic Equation Solution Dependence on Parameters.- 5. Stochastic Flow.- 4. Stochastic Equations on Smooth Manifolds.- 1. Stochastic Differentials.- 1.1. Ito’s Bundle.- 1.2. Stochastic Differentials on Manifolds.- 2. Stochastic Differential Equations on Manifolds.- 3. Stochastic Equations in Vector Bundles.- 3.1. Stochastic Equations on a Vector Bundle Total Space.- 3.2. Smooth Properties of Stochastic Equation Solutions.- 5. Kolmogorov Equations.- 1. Backward Kolmogorov Equations.- 1.1. General Arguments.- 1.2. Calculation of the Infinitesimal Operator of the Evolution Family Generated by a Random Process in a Linear Space.- 1.3. Infinitesimal Operators of Evolution Families Generated by a Manifold Valued Stochastic Process.- 1.4. Cauchy Problem for a Parabolic Equation.- 2. Quasilinear Parabolic Equations.- 2.1. General Approach.- 2.2. Local Manifolds.- 2.3. Smooth Solutions of Quasilinear Parabolic Equations.- 2.4. Cauchy Problem for Quasilinear Parabolic Equations over Manifold and Vector Bundles.- 3. Forward Kolmogorov Equations.- 3.1. Evolution Families in the Space of Measures.- 3.2. Smoothness Property of Transition Probability.- 6. Diffusion Processes on Lie Groups and Principal Fibre Bundles.- 1. Lie Groups and Smooth Bundles.- 1.1. Lie Groups and Lie Algebras.- 1.2. Transformation Groups and Fibre Bundles.- 1.3. Connections on Principal Fibre Bundles.- 1.4. Linear Connection on the Total Space of a Fibre Bundle.- 2. Invariant Stochastic Equations.- 2.1. Equations Invariant Under One-Parameter Group Actions.- 2.2. Stochastic Equations on a Lie Group.- 2.3. Stochastic Equations on Principal Fibre Bundles.- 2.4. Evolution Families in Sections of Principal and Associated Bundles.- 3. Stochastic Equations on Manifolds and their Solution Distribution Properties.- 3.1. Forward and Backward Derivatives and their Connections.- 3.2. Stochastic Equations on Lie Groups and the Smoothness Property of their Solution Distributions.- 3.3. Absolutely, Continuous Smooth Measures.- 3.4. Absolutely Continuous Stochastic Equation Solution Distributions.- 3.5. Admissible Transformations of Smooth Measures.- Historical Comments.- References.