Belopolskaya / Dalecky Stochastic Equations and Differential Geometry

1990
ISBN: 978-94-009-2215-0
Verlag: Springer Netherland
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 30, 260 Seiten, eBook

Reihe: Mathematics and its Applications

Zielgruppe

Research

Autoren/Hrsg.

Belopolskaya, Ya.I.

Dalecky, Yu.L.

Inhaltsverzeichnis

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.